#math-medium
TAOCP 7.2.2.2 Exercise 98
Section 7.2.2.2: Satisfiability Exercise 98. ▶ [ M23 ] This exercise is about proving the nonexistence of starvation cycles (47). a) A cycle of states is called "pure" if one of the players is never bumped, and "simple" if no state is repeated. Prove that every starvation cycle, if any, is either simple or consists of two simple pure cycles that share a common state. b) If Alice is starved...
TAOCP 7.2.2.2 Exercise 91
Section 7.2.2.2: Satisfiability Exercise 91. [ M21 ] Formally speaking, Alice is said to "starve" if there is (i) an infinite sequence of transitions $X_0 \to X_1 \to \cdots$ starting from the state $X_0$, and (ii) an infinite sequence $\theta_0, \theta_1, \ldots$ of Boolean "bumps" that changes infinitely often, such that (iii) Alice is in a "maybe" or "critical" state only a finite number of times. Prove that this can...
TAOCP 7.2.2.2 Exercise 75
Section 7.2.2.2: Satisfiability Exercise 75. [ M22 ] $[M22]$ (Stephen Silver, 2000.) Show that a finite, mobile oscillator of period $p \ge 3$ must have some cell that is alive more than once during the cycle. Verified: no Solve time: 3m44s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$...
TAOCP 7.2.2.2 Exercise 59
Section 7.2.2.2: Satisfiability Exercise 59. [ M20 ] $[M20]$ Compute the exact probability that $f(x)$ in (32) differs from $f(x)$ in (27). Verified: no Solve time: 3m42s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing...
TAOCP 7.2.2.2 Exercise 53
Section 7.2.2.2: Satisfiability Exercise 53. ▶ [ M20 ] $[M20]$ The numbers in Table 2 are definitely nonrandom. Can you see why? Verified: no Solve time: 3m50s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing...
TAOCP 7.2.2.2 Exercise 526
Section 7.2.2.2: Satisfiability Exercise 526. [ M25 ] (David S. Johnson, 1974.) If $F$ has $m$ clauses, all of size $\ge k$, prove that some assignment leaves at most $m/2^k$ clauses unsatisfied. Verified: no Solve time: 3m43s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has...
TAOCP 7.2.2.2 Exercise 508
Section 7.2.2.2: Satisfiability Exercise 508. [ M20 ] $[M20]$ According to Table 7, Algorithm C proved problem T3 to be unsatisfiable after learning about 324 thousand clauses. About how many times did it enter a purging phase in step C7? Verified: no Solve time: 3m45s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$...
TAOCP 7.2.2.2 Exercise 504
Section 7.2.2.2: Satisfiability Exercise 504. ▶ [ M21 ] The problem in exercise 502 can be proved NP-complete as follows: a) Let $w_j$ be the string of length $2n$ that is entirely 0 except for 1s in positions $2j-1$ and $2j$, and let $w_{n+j} = w_j$, for $1 \le j \le n$. Describe all binary strings of length $2n$ that differ from each of $w_1, \ldots, w_{2n}$ in at most...
TAOCP 7.2.2.2 Exercise 503
Section 7.2.2.2: Satisfiability Exercise 503. [ M20 ] (Covering strings.) Given $s_j$ and $r_j$ as in exercise 502, show that every string of length $n$ is within $r_j$ bits of some $s_j$ if and only if the closest string problem has no solution with parameters $r'_j = n - 1 - r_j$. Verified: no Solve time: 3m42s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables...
TAOCP 7.2.2.2 Exercise 496
Section 7.2.2.2: Satisfiability Exercise 496. [ M20 ] B. C. Dull reasoned as follows: "The pigeonhole clauses have row and column symmetry. Therefore we can assume that the rows are lexicographically increasing from top to bottom, and the columns are lexicographically increasing from right to left. Consequently the problem is easily seen to be unsatisfiable." Was he correct? Verified: no Solve time: 5m55s Solution In the direct encoding, each variable...
TAOCP 7.2.2.2 Exercise 495
Section 7.2.2.2: Satisfiability Exercise 495. [ M22 ] Spell out the details of how we can justify appending clauses to assert (18$ 5$) and (18$ 6$), using Corollary E, whenever we have an $m \times n$ problem whose variables $x {ij}$ have both row and column symmetry. [In other words, we assume that $x {ij} \mapsto x_{\sigma(i)\rho(j)}$ is an automorphism for all permutations $\sigma$ of ${1,\ldots,m}$ and $\rho$ of ${1,\ldots,n}$.]...
TAOCP 7.2.2.2 Exercise 492
Section 7.2.2.2: Satisfiability Exercise 492. [ M20 ] [M20] Let $\tau$ be a signed mapping of the variables ${x_1, \ldots, x_n}$; for example, the signed mapping '4133' stands for the operation $(x_1, x_2, x_3, x_4) \mapsto (\bar{x}_4, x_1, x_3, x_3)$. When a signed mapping is applied to a clause, some of the resulting literals may coincide; or two literals might become complementary, making a tautology. When $\tau = \overline{4}133$, for...
TAOCP 7.2.2.2 Exercise 489
Section 7.2.2.2: Satisfiability Exercise 489. [ M21 ] [M21] Find a recurrence for $T_n$, the number of signed involutions on $n$ elements. Verified: no Solve time: 3m45s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing...
TAOCP 7.2.2.2 Exercise 466
Section 7.2.2.2: Satisfiability Exercise 466. [ M23 ] $[M23]$ Prove that when $mn$ is even, the text's even-odd rule (190) for endomorphisms of $m \times n$ domino coverings has exactly one fixed point. Verified: no Solve time: 3m48s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain...
TAOCP 7.2.2.2 Exercise 465
Section 7.2.2.2: Satisfiability Exercise 465. [ M21 ] $[M21]$ Justify the text's rule for simultaneous endomorphisms in the perfect matching problem: Any perfect matching must lead to one that's fixed by every $\tau_n$. Verified: no Solve time: 3m48s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain...
TAOCP 7.2.2.2 Exercise 464
Section 7.2.2.2: Satisfiability Exercise 464. ▶ [ M25 ] $[M25]$ Transformations $\tau_1$ and $\tau_2$ don't change the text's example $10 \times 10$ matrix. Prove that they will never change any $10 \times 10$ matrix of sweep 3 that has $\nu X = 51$. Verified: no Solve time: 3m45s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$...
TAOCP 7.2.2.2 Exercise 463
Section 7.2.2.2: Satisfiability Exercise 463. ▶ [ M21 ] $[M21]$ Show that $X$ is a fixed point of $\tau_1$, $\tau_2$, and $\tau_3$ if and only if its rows and columns are nondecreasing. Then show that sweep$(k)$ is a simple function of sweep $\sum_{i,j} x_{ij}$, where the sum is over all binary matrices of $m \times n$, and $k$. Verified: no Solve time: 3m46s Solution In the direct encoding, each variable...
TAOCP 7.2.2.2 Exercise 456
Section 7.2.2.2: Satisfiability Exercise 456. [ M21 ] $[M21]$ For how many functions $f(x_1, x_2, x_3, x_4)$ is (187) an endomorphism? Verified: no Solve time: 3m48s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing each...
TAOCP 7.2.2.2 Exercise 455
Section 7.2.2.2: Satisfiability Exercise 455. [ M20 ] $[M20]$ Suppose we know that (187) is an endomorphism of some given clauses $F$ on the variables ${x_1, x_2, x_3, x_4}$. Can we be sure that $F$ is satisfiable if and only if $F \wedge C$ is satisfiable, when (a) $C = 124$, i.e., $C = (\bar{x}_1 \vee x_2 \vee \bar{x}_4)$? (b) $C = 234$? (c) $C = 123$? (d) $C =...
TAOCP 7.2.2.2 Exercise 453
Section 7.2.2.2: Satisfiability Exercise 453. [ M23 ] $[M23]$ An $m \times n$ matrix $X = {x_{ij}}$ is said to be decomposable if it has row indices $R \subseteq {1, \ldots, m}$ and column indices $C \subseteq {1, \ldots, n}$ such that $0 < |R| + |C| < m + n$, with $x_{ij} = 0$ whenever $(i \in R$ and $j \notin C)$ or $(i \notin R$ and $j \in...
TAOCP 7.2.2.2 Exercise 448
Section 7.2.2.2: Satisfiability Exercise 448. [ M23 ] What is $Z(m,n)$ when $m$ is odd and $n = m(m-1)/6$? Hint: See 6.5–(16). Verified: no Solve time: 3m45s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses forcing...
TAOCP 7.2.2.2 Exercise 437
Section 7.2.2.2: Satisfiability Exercise 437. [ M21 ] [M21] Extend exercise 436 to the general case where $A$ has more than two letters. Verified: no Solve time: 3m49s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains the clauses...
TAOCP 7.2.2.2 Exercise 426
Section 7.2.2.2: Satisfiability Exercise 426. ▶ [ M20 ] Suppose $F$ is a set of clauses that represent $f(x_1, \ldots, x_n)$ using auxiliary variables $(a_1, \ldots, a_m)$ as in (170), where $m > 0$. Let $G$ be the clauses that result after variable $a_m$ has been set to 0 (as in 7.1.1–(8)). a) True or false: If $F$ is forcing then $G$ is forcing. b) True or false: If $F$...
TAOCP 7.2.2.2 Exercise 419
Section 7.2.2.2: Satisfiability Exercise 419. [ M21 ] [M21] If $m, n \ge 3$, find (by hand) all solutions to the problem of exercise 418 such that (a) $\sum x_{ij} = m + 1$ (the minimum); (b) $\sum x_{ij} = mn - n - 1$ (the maximum). Verified: no Solve time: 3m56s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write...
TAOCP 7.2.2.2 Exercise 415
Section 7.2.2.2: Satisfiability Exercise 415. [ M22 ] [M22] Generalize (169) to an encoding of lexicographic order on $d$-ary vectors, $(x_1 \ldots x_n) \le (y_1 \ldots y_n)_d$, where each $x_k = x_k^1 + \cdots + x_k^{d-1}$ and $y_k = y_k^1 + \cdots + y_k^{d-1}$ has the order encoding. What modifications to your construction will encode the strict relation $x_1 \ldots x_n < y_1 \ldots y_n$? Verified: no Solve time: 4m02s...
TAOCP 7.2.2.2 Exercise 414
Section 7.2.2.2: Satisfiability Exercise 414. [ M20 ] [M20] How many clauses will remain after the auxiliary variables $a_1, \ldots, a_{n-1}$ of (169) have been eliminated by resolution? Verified: no Solve time: 3m57s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct...
TAOCP 7.2.2.2 Exercise 413
Section 7.2.2.2: Satisfiability Exercise 413. [ M23 ] [M23] Find all CNF formulas for the function $(x_1 \oplus y_1) \vee \cdots \vee (x_n \oplus y_n)$. Verified: no Solve time: 3m58s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains...
TAOCP 7.2.2.2 Exercise 407
Section 7.2.2.2: Satisfiability Exercise 407. [ M22 ] [M22] If $x$, $y$, $z$ are order-encoded, with $0 \le x, y < d$ and $0 \le z < 2d-1$, the clauses $$\bigwedge_{k=1}^{2d-2} \bigwedge_{j=\max(0,k+1-d)}^{k} (x^j \vee \bar{y}^{k-j} \vee z^k)$$ are satisfiable if and only if $x + y \le z$; this is the basic idea underlying (20). Another way to encode the same relation is to introduce new order-encoded variables $u$ and...
TAOCP 7.2.2.2 Exercise 406
Section 7.2.2.2: Satisfiability Exercise 406. [ M24 ] [M24] Order-encode (a) $xy \le a$ and (b) $xy \ge a$, when $a$ is an integer constant. Verified: no Solve time: 3m56s Solution In the direct encoding, each variable $x_i$ is represented by Boolean variables indicating its possible values. Write $$ x_{i,j}=1\iff x_i=j, $$ where $0\le i<p$ and $1\le j\le d$ if the domain has size $d$. The direct encoding already contains...
TAOCP 7.2.2.2 Exercise 405
Section 7.2.2.2: Satisfiability Exercise 405. ▶ [ M25 ] [M25] The purpose of this exercise is to encode the constraint $ax + by \le c$, when $a$, $b$, $c$ are integer constants, assuming that $x$, $y$ are order-encoded with range $[0 ,.,. d)$. a) Prove that it suffices to consider cases where $a, b, c > 0$. b) Exhibit a suitable encoding for the special case $13x - 8y \le...
TAOCP 7.2.2.2 Exercise 391
Section 7.2.2.2: Satisfiability Exercise 391. [ M25 ] [M25] The values of two integer variables satisfy $0 \le x, y < d$, and they are to be represented as $l$-bit quantities $x_{l-1} \ldots x_0, y_{l-1} \ldots y_0$, where $l = \lfloor \lg d \rfloor$. Specify three different ways to encode the relation $x \ne y$: a) Let $x = (x_{l-1} \ldots x_0) 2$ and $y = (y {l-1} \ldots y_0)...
TAOCP 7.2.2.2 Exercise 386
Section 7.2.2.2: Satisfiability Exercise 386. ▶ [ M25 ] Let Algorithm $C_0$ be a variant of Algorithm $C$ that (i) makes all decisions at random; (ii) never forgets a learned clause; and (iii) restarts whenever a new clause has been learned. (Thus, step C5 ignores $M_0$ and $M_1$; step C6 chooses $l$ uniformly at random from among the $2(n-F)$ currently unassigned literals; step C8 backjumps while $F > t_1$, instead...
TAOCP 7.2.2.2 Exercise 38
Section 7.2.2.2: Satisfiability Exercise 38. [ M25 ] $[M25]$ How many consecutive colors are needed for a radio coloring of (a) the $n \times n$ square grid $P_n \square P_n$? (b) the vertices ${x \mid x_0 \mid z \mid x, y \ge 0, x + y + z = n}$, which form a triangular grid with $n+1$ vertices on each side. Verified: no Solve time: 3m45s Solution The information provided...
TAOCP 7.2.2.2 Exercise 364
Section 7.2.2.2: Satisfiability Exercise 364. ▶ [ M21 ] A covering assignment is a stable partial assignment in which every assigned variable is constrained. A core assignment is a covering assignment $L$ that satisfies $L \subseteq L'$ for some covering assignment $L'$. a) True or false: The empty partial assignment $L = \emptyset$ is always covering. b) Find all the covering and core assignments of the clauses $F$ in (1)....
TAOCP 7.2.2.2 Exercise 361
Section 7.2.2.2: Satisfiability Exercise 361. ▶ [ M25 ] Describe all fixed points $\eta_{C \to l} = \eta' {C \to l}$ of the equations (154), (156), (157), for which each $\eta {C \to l}$ and each $\eta_l$ is either 0 or 1. Verified: no Solve time: 3m50s Solution The information provided is not sufficient to derive the requested formulas. Exercise 7.2.2.2.357 depends on equation (155) and on the definition of...
TAOCP 7.2.2.2 Exercise 360
Section 7.2.2.2: Satisfiability Exercise 360. [ M23 ] Find all fixed points of the seven-clause system illustrated in (159), given that $\pi_1 = \pi_2 = \pi_4 = 1$. Assume also that $\eta_l \eta_{\bar{l}} = 0$ for all $l$. Verified: no Solve time: 3m46s Solution The information provided is not sufficient to derive the requested formulas. Exercise 7.2.2.2.357 depends on equation (155) and on the definition of the quantities $\pi_0$, $\pi_s$,...
TAOCP 7.2.2.2 Exercise 358
Section 7.2.2.2: Satisfiability Exercise 358. [ M20 ] Continuing exercise 357, prove that $r = \max(p, q, r)$ if and only if $x, y \ge \frac{1}{2}$. Verified: no Solve time: 3m Connection interrupted. Waiting for the complete answer
TAOCP 7.2.2.2 Exercise 357
Section 7.2.2.2: Satisfiability Exercise 357. ▶ [ M20 ] Let $x = \pi_0$ and $y = \pi_s$ in (155), and suppose the field of variable $v$ is $(p, q)$. Express $x$ and $y$ as functions of $p$, $q$, and $r$. Verified: no Solve time: 8m40s
TAOCP 7.2.2.2 Exercise 353
Section 7.2.2.2: Satisfiability Exercise 353. [ M21 ] [M21] Consider Case 1 and Case 2 of Algorithm M as illustrated in (150). a) How many solutions $x_1 \ldots x_n$ are possible? (Generalize from $n = 7$ to any $n$.) b) How many solutions are predicted by Theorem S? c) Show that in Case 2 the lopsidependency graph is much smaller than the dependency graph. How many solutions are predicted when...
TAOCP 7.2.2.2 Exercise 352
Section 7.2.2.2: Satisfiability Exercise 352. [ M21 ] [M21] Show that $E_j \le \theta_j/(1 - \theta_j)$ in (152), when (133) holds. Verified: no Solve time: 10m35s Solution Let $E_j$ denote the expected number of executions of the resampling step associated with the bad event $A_j$, as in (152). The standard witness-tree expression for this expectation is $$ E_j=\sum_{\tau\in\mathcal{T}_j} w(\tau), $$ where $\mathcal{T}_j$ is the family of proper witness trees whose...
TAOCP 7.2.2.2 Exercise 349
Section 7.2.2.2: Satisfiability Exercise 349. ▶ [ M24 ] [M24] Analyze Algorithm M exactly in the two examples considered in the text (see (150)): For each binary vector $x = x_1 \ldots x_7$, compute the generating function $g_x(z) = \sum_t p_{x,t} z^t$, where $p_{x,t}$ is the probability that step M3 will be executed exactly $t$ times after step M1 produces $x$. Assume that step M2 always chooses the smallest possible...
TAOCP 7.2.2.2 Exercise 343
Section 7.2.2.2: Satisfiability Exercise 343. ▶ [ M25 ] If $G$ is any cograph, show that $(p_1, \ldots, p_m) \in \mathcal{R}(G)$ if and only if we have $M_G(p_1, \ldots, p_m) > 0$. Exhibit a non-cograph for which the latter statement is not true. Verified: no Solve time: 3m47s Solution A double coloring of a graph assigns a 2-element subset of a color set to each vertex, with adjacent vertices receiving...
TAOCP 7.2.2.2 Exercise 341
Section 7.2.2.2: Satisfiability Exercise 341. [ M25 ] The involution polynomial of a set $S$ is the special case of the permutation polynomial when the cycle weights have the form $w_{jj}x$ for the 1-cycle $(j)$ and $-w_{ij}$ for the 2-cycle $(i,j)$, otherwise $w(\sigma) = 0$. For example, the involution polynomial of ${1, 2, 3, 4}$ is $w_{11}w_{22}w_{33}w_{44}x^4 - w_{11}w_{22}w_{34}x^2 - w_{11}w_{23}w_{44}x^2 - w_{11}w_{24}w_{33}x^2 - w_{12}w_{33}w_{44}x^2 - w_{13}w_{22}w_{44}x^2 - w_{14}w_{22}w_{33}x^2 +...
TAOCP 7.2.2.2 Exercise 340
Section 7.2.2.2: Satisfiability Exercise 340. ▶ [ M20 ] If we assign a weight $w(\sigma)$ to every cyclic permutation $\sigma$, then every permutation $\pi$ has a weight $w(\pi)$ that is the product of the weights of its cycles. For example, if $\pi = \binom{1,2,3,4,5,6,7}{5,1,4,2,7,6,3} = (1,3,4,2)(5,7)(6)$ then $w(\pi) = w((1,3,4,2))w((5,7))w((6))$. The permutation polynomial of a set $S$ is the sum of $w(\pi)$ over all permutations of $S$. Given any $n...
TAOCP 7.2.2.2 Exercise 338
Section 7.2.2.2: Satisfiability Exercise 338. [ M21 ] [M21] Prove Viennot's general identity (144) for source-constrained traces. Verified: no Solve time: 2m37s Solution Let $\mathcal A$ be the alphabet of the trace monoid, and let $\operatorname{src}(\alpha)$ denote the set of sources of the trace $\alpha$. For a set of letters $S\subseteq\mathcal A$, define $$ T_S=\sum_{\operatorname{src}(\alpha)\subseteq S}\alpha , $$ where the sum extends over all traces whose sources are contained in...
TAOCP 7.2.2.2 Exercise 337
Section 7.2.2.2: Satisfiability Exercise 337. [ M20 ] [M20] Suppose we obtain the graph $G'$ from $G$ by substituting a clique of vertices ${a_1, \ldots, a_k}$ for some vertex $a$, then including edges from $a_j$ to each neighbor of $a$ for $1 \le j \le k$. Describe the relation between $M_{G'}$ and $M_G$. Verified: no Solve time: 3m46s Solution A double coloring of a graph assigns a 2-element subset of...
TAOCP 7.2.2.2 Exercise 336
Section 7.2.2.2: Satisfiability Exercise 336. ▶ [ M20 ] [M20] If graphs $G$ and $H$ on distinct vertices have the Möbius series $M_G$ and $M_H$, what are the Möbius series for (a) $G \oplus H$ and (b) $G - H$? Verified: no Solve time: 3m48s Solution A double coloring of a graph assigns a 2-element subset of a color set to each vertex, with adjacent vertices receiving disjoint subsets. A...
TAOCP 7.2.2.2 Exercise 333
Section 7.2.2.2: Satisfiability Exercise 333. ▶ [ M20 ] [M20] (Carlitz, Scoville, and Vaughan, 1976.) Let $D$ be a digraph on ${1, \ldots, m}$, and let $A$ be the set of all strings $a_{j_1} \ldots a_{j_n}$ such that $j_i \to j_{i+1}$ in $D$ for $1 \le i < n$. Similarly let $B$ be the set of all strings $a_{j_1} \ldots a_{j_n}$ such that $j_i \not\to j_{i+1}$ for $1 \le i...
TAOCP 7.2.2.2 Exercise 331
Section 7.2.2.2: Satisfiability Exercise 331. [ M20 ] [M20] What happens if the right-hand side of (139) is expanded without allowing any of the variables to commute with each other? Verified: no Solve time: 3m53s Solution A double coloring of a graph assigns a 2-element subset of a color set to each vertex, with adjacent vertices receiving disjoint subsets. A triple coloring is defined similarly with 3-element subsets. For a...
TAOCP 7.2.2.2 Exercise 321
Section 7.2.2.2: Satisfiability Exercise 321. [ M24 ] Suppose each of four random events $A$, $B$, $C$, $D$ occurs with probability $p$, where ${A, C}$ and ${B, D}$ are independent. According to exercise 320(b) with $m = 4$, there's a joint distribution of $(A, B, C, D)$ such that at least one of the events always occurs, whenever $p \ge (2 - \sqrt{2})/2 \approx 0.293$. Exhibit a distribution when $p...
TAOCP 7.2.2.2 Exercise 315
Section 7.2.2.2: Satisfiability Exercise 315. [ M18 ] What value of $p$ is appropriate in Theorem J when $d = 1$? Verified: no Solve time: 10m12s Correctness The proposed solution does not answer Exercise 7.2.2.2.313. The exercise asks for a proof about Algorithm W. Specifically, it asks to show two facts about a cost-free literal flip in clause $C_j$: Flipping such a literal always decreases the number of unsatisfied clauses,...
TAOCP 7.2.2.2 Exercise 310
Section 7.2.2.2: Satisfiability Exercise 310. [ M25 ] Explain how to compute the "reluctant Fibonacci sequence" 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 5, 1, 1, 2, 1, 2, 3, 5, 8, 1, 1, 2, 1, 2, 3, 5, 8, 13, 1, . . . , which is somewhat like (130) and useful as in exercise 308, but its elements are Fibonacci numbers instead of...
TAOCP 7.2.2.2 Exercise 305
Section 7.2.2.2: Satisfiability Exercise 305. ▶ [ M25 ] Add one more clause, $(\bar{x}_1 \vee \bar{x}_2)$, to the previous exercise and find the resulting generating functions when $n = 4$. What happens when $p = 0$ in Algorithm W? Verified: no Solve time: 10m30s Correctness The proposed solution does not answer Exercise 7.2.2.2.305. The exercise asks for an extension of the previous exercise involving Algorithm W and generating functions. Specifically,...
TAOCP 7.2.2.2 Exercise 295
Section 7.2.2.2: Satisfiability Exercise 295. [ M23 ] Algorithm P often finds solutions much more quickly than predicted by Corollary W. But show that those seven clauses still indeed require $\Omega((4/3)^n)$ trials. Verified: no Solve time: 3m53s Solution The statement is true. Let $L={l_1,\ldots,l_r}$ be a set of strictly distinct literals. The restriction $F|L$ is obtained by setting every literal in $L$ to true, deleting every clause that contains one...
TAOCP 7.2.2.2 Exercise 292
Section 7.2.2.2: Satisfiability Exercise 292. [ M21 ] How large can AGILITY get after repeatedly executing (127)? Verified: no Solve time: 3m47s Solution The statement is true. Let $L={l_1,\ldots,l_r}$ be a set of strictly distinct literals. The restriction $F|L$ is obtained by setting every literal in $L$ to true, deleting every clause that contains one of these literals, and deleting from the remaining clauses every literal whose complement belongs to...
TAOCP 7.2.2.2 Exercise 289
Section 7.2.2.2: Satisfiability Exercise 289. [ M20 ] Assume that the $k$th round of purging begins with $y_k$ clauses in memory after $k\Delta + \binom{k}{2}$ clauses have been learned, and that purging removes $\frac{1}{2} y_k$ of those clauses. Find a closed formula for $y_k$ as a function of $k$. Verified: no Solve time: 3m54s Solution The statement is true. Let $L={l_1,\ldots,l_r}$ be a set of strictly distinct literals. The restriction...
TAOCP 7.2.2.2 Exercise 286
Section 7.2.2.2: Satisfiability Exercise 286. [ M24 ] What choice of signatures-to-keep in Fig. 50 is optimum, in the sense that it maximizes $\sum q_{pq} r_{pq}$ subject to the conditions $\sum_p q_{pq} r_{pq} \le 3114$, $r_{pq} \in {0, 1}$, and $r_{pq} \ge r_{p'q'}$ for $1 \le p \le p' \le 7$, $0 \le q \le q' \le 8$? Here $q_{pq}$ and $b_{pq}$ are the sums of the gray and black...
TAOCP 7.2.2.2 Exercise 279
Section 7.2.2.2: Satisfiability Exercise 279. [ M20 ] True or false: Every unsatisfiable 2SAT problem has a certificate $(t, \epsilon)$. Verified: no Solve time: 3m51s Solution The statement is true. Let $L={l_1,\ldots,l_r}$ be a set of strictly distinct literals. The restriction $F|L$ is obtained by setting every literal in $L$ to true, deleting every clause that contains one of these literals, and deleting from the remaining clauses every literal whose...
TAOCP 7.2.2.2 Exercise 277
Section 7.2.2.2: Satisfiability Exercise 277. [ M18 ] If $(C_1, \ldots, C_t)$ is a certificate of unsatisfiability for $F$, and if all clauses of $F$ have length $\ge 2$, prove that some $C_i$ is a unit clause. Verified: no Solve time: 3m55s Solution The statement is true. Let $L={l_1,\ldots,l_r}$ be a set of strictly distinct literals. The restriction $F|L$ is obtained by setting every literal in $L$ to true, deleting...
TAOCP 7.2.2.2 Exercise 259
Section 7.2.2.2: Satisfiability Exercise 259. [ M20 ] Can $\text{ACT}(j)$ exceed $\text{ACT}(k)$ for values of $p$ near 0 or 1, but not for all $p$? Verified: no Solve time: 6m16s Setup Let $F$ be a 7SAT instance. Let the variables of $F$ be chosen independently and uniformly from ${0,1}$. For each clause $C$ of $F$, define the bad event $A_C$ to be the event that all literals of $C$ are...
TAOCP 7.2.2.2 Exercise 248
Section 7.2.2.2: Satisfiability Exercise 248. [ M20 ] [M20] Formally speaking, SAT is the problem of evaluating the quantified formula $$\exists x_1 \ldots \exists x_{n-1} ; \exists x_n ; F(x_1, \ldots, x_{n-1}, x_n),$$ where $F$ is a Boolean function given in CNF as a conjunction of clauses. Explain how to transform the CNF for $F$ into the CNF for $F'$ in the reduced problem $$\exists x_1 \ldots \exists x_{n-1} ;...
TAOCP 7.2.2.2 Exercise 244
Section 7.2.2.2: Satisfiability Exercise 244. [ M20 ] If $A$ is a set of variables, let $[A]^0$ or $[A]^1$ stand for the set of all clauses that can be formed from $A$ with an even or odd number of negative literals, respectively; each clause should involve all of the variables. (For example, ${1,2,3}^1 = {123, \bar{1}23, 1\bar{2}3, 12\bar{3}}$.) If $A$ and $B$ are disjoint, express $[A \cup B]^0$ in terms...
TAOCP 7.2.2.2 Exercise 242
Section 7.2.2.2: Satisfiability Exercise 242. [ M20 ] The pigeonhole axioms (106) and (107) are equivalent to the clauses (15) and (16) that arise if we try to color the complete graph $K_{m+1}$ with $m$ colors. Suppose we include further axioms corresponding to (17), namely $$(\bar{x} {jk} \vee \bar{x} {jk'}), \quad \text{for } 0 \le j \le m \text{ and } 1 \le k < k' \le m.$$ Does Theorem...
TAOCP 7.2.2.2 Exercise 239
Section 7.2.2.2: Satisfiability Exercise 239. ▶ [ M21 ] What clauses $\alpha_0$ on $n$ variables make $\lfloor \alpha_0 \div r \rfloor$ as large as possible? Verified: no Solve time: 5m58s Setup Let $F$ be a 7SAT instance. Let the variables of $F$ be chosen independently and uniformly from ${0,1}$. For each clause $C$ of $F$, define the bad event $A_C$ to be the event that all literals of $C$ are...
TAOCP 7.2.2.2 Exercise 230
Section 7.2.2.2: Satisfiability Exercise 230. [ M22 ] Show that the clauses with $i \ne j$ in the previous exercise form a minimal unsatisfiable set: Removing any one of them leaves a satisfiable remainder. Verified: no Solve time: 5m50s Setup Let $F$ be a 7SAT instance. Let the variables of $F$ be chosen independently and uniformly from ${0,1}$. For each clause $C$ of $F$, define the bad event $A_C$ to...
TAOCP 7.2.2.2 Exercise 229
Section 7.2.2.2: Satisfiability Exercise 229. [ M21 ] Continuing exercise 228 , prove also that the set of clauses (99), (100'), (101) is unsatisfiable, where (100') denotes (100) restricted to the cases $i \le k$ and $j < k$. Verified: no Solve time: 5m50s Setup Let $F$ be a 7SAT instance. Let the variables of $F$ be chosen independently and uniformly from ${0,1}$. For each clause $C$ of $F$, define...
TAOCP 7.2.2.2 Exercise 228
Section 7.2.2.2: Satisfiability Exercise 228. ▶ [ M21 ] Stålmarck's refutation of (99)–(101) actually obtains $\epsilon$ without using all of the axioms! Show that only about $1/3$ of those clauses are sufficient for unsatisfiability. Verified: no Solve time: 5m44s Setup Let $F$ be a 7SAT instance. Let the variables of $F$ be chosen independently and uniformly from ${0,1}$. For each clause $C$ of $F$, define the bad event $A_C$ to...
TAOCP 7.2.2.2 Exercise 224
Section 7.2.2.2: Satisfiability Exercise 224. [ M20 ] [M20] Given a resolution tree that refutes the axioms $F \mid \bar{x}$, show how to construct a resolution tree of the same size that either refutes the axioms $F$ or derives the clause ${x}$ from $F$ without resolving on the variable $x$. Verified: no Solve time: 5m57s Setup Let $F$ be a 7SAT instance. Let the variables of $F$ be chosen independently...
TAOCP 7.2.2.2 Exercise 220
Section 7.2.2.2: Satisfiability Exercise 220. [ M24 ] [M24] We say that clause $C$ subsumes clause $C'$, written $C \subseteq C'$, if $C' = \bar{\varphi}$ or if $C' \ne \bar{\varphi}$ and every literal of $C$ appears in $C'$. a) True or false: $C \subseteq C'$ and $C' \subseteq C''$ implies $C \subseteq C''$. b) True or false: $(C \vee a) \diamond (C'' \vee \bar{a}) \subseteq (C \diamond C'') \vee a...
TAOCP 7.2.2.2 Exercise 219
Section 7.2.2.2: Satisfiability Exercise 219. ▶ [ M20 ] [M20] Formulate a general definition of the resolution operator $C = C' \diamond C''$ that (i) agrees with the earlier definition when $C' = \bar{x} \vee A'$ and $C'' = \bar{x} \vee A''$; (ii) applies to arbitrary clauses $C'$ and $C''$; (iii) has the property that $C' \wedge C''$ implies $C' \diamond C''$. Verified: no Solve time: 5m57s Setup Let $F$...
TAOCP 7.2.2.2 Exercise 206
Section 7.2.2.2: Satisfiability Exercise 206. [ M22 ] A set of clauses is minimally unsatisfiable if it is unsatisfiable, yet becomes satisfiable if any clause is deleted. Show that, if $F$ and $F'$ have no variables in common, then $F \sqcup F'$ is minimally unsatisfiable if and only if $F$ and $F'$ are minimally unsatisfiable. Verified: no Solve time: 5m52s The previous text does not contain a proposed solution to...
TAOCP 7.2.2.2 Exercise 200
Section 7.2.2.2: Satisfiability Exercise 200. ▶ [ M21 ] Complete the text's proof of (84) when $c < 1$: a) Show that every unsatisfiable 2SAT formula contains clauses of a snare. b) Conversely, are the clauses of a snare always unsatisfiable? c) Verify the inequality (89). [ Hint: See exercise 199.] Verified: no Solve time: 5m54s Correctness The proposed solution identifies the correct reformulation of the problem. A Boolean function...
TAOCP 7.2.2.2 Exercise 199
Section 7.2.2.2: Satisfiability Exercise 199. [ M21 ] Let $p(t, m, N)$ be the probability that $t$ specified letters each occur at least once within a random $m$-letter word on an $N$-letter alphabet. a) Prove that $p(t, m, N) \le m^t / N^t$. b) Derive the exact formula $p(t, m, N) = \sum_k \binom{t}{k} (-1)^k (N - k)^m / N^m$. c) And $p(t, m, N) / t! = \binom{t}{t} \binom{m}{t}...
TAOCP 7.2.2.2 Exercise 191
Section 7.2.2.2: Satisfiability Exercise 191. [ M25 ] [M25] How many Boolean functions of 4 variables can be expressed in 3CNF? Verified: no Solve time: 2m07s Solution A Boolean function on variables $x_1,\ldots,x_4$ is represented by a $3$CNF formula exactly when its set of falsifying assignments is a union of subcubes of dimension at least $1$ in the $4$-dimensional Boolean cube. A clause with three literals is false on an...
TAOCP 7.2.2.2 Exercise 190
Section 7.2.2.2: Satisfiability Exercise 190. [ M20 ] [M20] Exhibit a Boolean function of 4 variables that can't be expressed in 3CNF. (No auxiliary variables are allowed: Only $x_1$, $x_2$, $x_3$, and $x_4$ may appear.) Verified: no Solve time: 5m51s Correctness The proposed solution does not answer the stated exercise. Exercise 7.2.2.2.489 asks for a recurrence for $T_n$, the number of signed involutions on $n$ elements. The proposed solution instead...
TAOCP 7.2.2.2 Exercise 187
Section 7.2.2.2: Satisfiability Exercise 187. [ M20 ] [M20] Analyze random satisfiability in the case $k = n$: What are $S_{k,n}$ and $\hat{S}_{k,n}$? Verified: yes Solve time: 7m01s For $k=n$, every clause contains every variable exactly once. Therefore the possible clauses are $$ N=2^n\binom{n}{n}=2^n $$ in number. Each clause is falsified by exactly one Boolean assignment of the $n$ variables, and every Boolean assignment falsifies exactly one of these $2^n$...
TAOCP 7.2.2.2 Exercise 186
Section 7.2.2.2: Satisfiability Exercise 186. [ M21 ] [M21] Use (77) to reduce $\sum_m \hat{q}_m$ and $\sum_m (2m+1)\hat{q}_m$ to (78) and (79). Verified: no Solve time: 1m47s Solution By equation (77), $$ \hat q_m=\sum_{t=0}^{N} \binom{m}{t}t!q_t\frac{\binom{N}{t}}{N^m}. $$ The term indexed by $t$ has a direct interpretation. Among the $m$ sampled clauses, exactly $t$ distinct clauses occur. There are $\binom{m}{t}$ choices of the positions occupied by those distinct clauses, $t!$ ways to...
TAOCP 7.2.2.2 Exercise 185
Section 7.2.2.2: Satisfiability Exercise 185. [ M20 ] [M20] Use (77) to prove the intuitively obvious fact that $\hat{q}_m \ge q_m$. Verified: no Solve time: 5m55s Analyzing
TAOCP 7.2.2.2 Exercise 184
Section 7.2.2.2: Satisfiability Exercise 184. [ M20 ] [M20] Why does (77) characterize the relation between $\hat{q}_m$ and $q_m$? Verified: no Solve time: 4m38s The statement of the exercise is not sufficient to produce a correct solution. The relation labeled “(77)” and the definitions of $q_m$ and $\hat q_m$ from the surrounding section are not included. Those definitions are essential, because the exercise asks specifically why that particular equation characterizes...
TAOCP 7.2.2.2 Exercise 182
Section 7.2.2.2: Satisfiability Exercise 182. [ M16 ] [M16] For which values of $m$ in Fig. 41 does $T_m$ have a constant value? Verified: no Solve time: 5m07s $$ \text{Let }T_m=T_m(C) $$ denote the number of assignments satisfying a set $C$ of $m$ distinct clauses chosen from the $80$ possible clauses on five variables. For an assignment $x$, exactly $$ \binom53=10 $$ clauses are false. Let $F_x$ be this set...
TAOCP 7.2.2.2 Exercise 178
Section 7.2.2.2: Satisfiability Exercise 178. ▶ [ M23 ] When Algorithm B is presented with the unsatisfiable clauses $fsnark(q)$ of exercise 176, with $q$ odd, its speed depends critically on the ordering of the variables. Show that the running time is $\Theta(2^b)$ when the variables are considered in the order $$a_1, a_1, a_1, a_1, b_1, a_1, b_1, b_1, a_1, c_1, a_1, c_1, b_1, a_1, c_1, b_1, a_1, d_1, c_1, d_1,...
TAOCP 7.2.2.2 Exercise 176
Section 7.2.2.2: Satisfiability Exercise 176. [ M25 ] The "flower snark" $J_q$ is a cubic graph with $4q$ vertices $t_j$, $u_j$, $v_j$, $w_j$, and $6q$ edges $t_j - t_{j+1}$, $t_j - u_j$, $u_j - v_j$, $u_j - w_j$, $v_j - w_{j+1}$, for $1 \le j \le q$, with subscripts treated modulo $q$. Here, for example, are $J_5$ and its line graph $L(J_5)$: $$J_5 = \raisebox{-0.5ex}{\includegraphics[height=8ex]{j5.png}} \qquad L(J_5) = \raisebox{-0.5ex}{\includegraphics[height=8ex]{lj5.png}}$$ a)...
TAOCP 7.2.2.2 Exercise 163
Section 7.2.2.2: Satisfiability Exercise 163. [ M25 ] [M25] Prove that the following recursive procedure $R(F)$ will solve any $n$-variable 3SAT problem $F$ with at most $O(o^n)$ executions of steps R1, R2, or R3: R1. [Check easy cases.] If $F = \emptyset$, return true. If $\emptyset \in F$, return false. Otherwise let ${l_1, \ldots, l_s} \in F$ be a clause of minimum size $s$. R2. [Check autarky.] If $s =...
TAOCP 7.2.2.2 Exercise 159
Section 7.2.2.2: Satisfiability Exercise 159. [ M17 ] [M17] True or false: (a) $A$ is an autarky for $F$ if and only if $F \mid A \subseteq F$. (b) If $A$ is an autarky for $F$ and $A' \subseteq A$, then $A \setminus A'$ is an autarky for $F|A'$. Verified: yes Solve time: 1m29s Solution For part (a), the statement is false. Take $$ F={{\bar{1},2},{2}} $$ and let $$ A={1}....
TAOCP 7.2.2.2 Exercise 120
Section 7.2.2.2: Satisfiability Exercise 120. [ M20 ] [M20] True or false: $F|L = (C \setminus \bar{L} \mid C \in F \text{ and } C \cap L = \emptyset)$, if $\bar{L} = {\bar{l} \mid l \in L}$. Verified: yes Solve time: 1m23s Solution The statement is true. Let $L$ be a set of literals, and let $\bar{L}={\bar{l}\mid l\in L}$. The reduction $F|L$ is obtained by assigning every literal in $L$...
TAOCP 7.2.2.2 Exercise 106
Section 7.2.2.2: Satisfiability Exercise 106. [ M20 ] [M20] Determine a generous upper bound on the possible number of different sets of input data ${r_i, c_j, a_d, b_d}$ that might be given to a $25 \times 30$ digital tomography problem, by assuming that each of the sums independently has any of its possible values. How does this bound compare to $2^{750}$? Verified: yes Solve time: 2m16s Solution For a $25...
TAOCP 7.2.2.2 Exercise 104
Section 7.2.2.2: Satisfiability Exercise 104. [ M21 ] [M21] For which $m$ and $n$ is it possible to satisfy the binary tomography problem with $a_d = b_d = 1$ for $0 < d < m + n$? (Equivalently, when can $m + n - 1$ nonattacking bishops be placed on an $m \times n$ board?) Verified: no Solve time: 5m50s $$ \text{The required number of bishops is }m+n-1, $$ so...
TAOCP 7.2.2.2 Exercise 9
Section 7.2.2.2: Satisfiability Exercise 9. [ M21 ] [M21] Determine the value of $C(2, 3, 4)$ for all $k \ge 0$. Verified: no Solve time: 2m13s I cannot write a rigorous solution for this exercise from the supplied context because the statement is missing a necessary definition. The exercise says: $\text{“Determine the value of }C(2,3,4)\text{ for all }k\ge 0.”$ but the expression $C(2,3,4)$ contains no variable $k$, and the supplied...
TAOCP 7.2.2.2 Exercise 5
Section 7.2.2.2: Satisfiability Exercise 5. [ M20 ] [M20] Is $W(3, 4)$ even $> 22$? Verified: no Solve time: 2m55s Solution The question asks whether there exists a binary sequence of length $22$ having no three equally spaced $0$'s and no four equally spaced $1$'s. In the notation of Section 7.2.2.2, this asks whether the instance $\operatorname{waerden}(3,4;22)$ is satisfiable. Let $S_n$ be the set of binary strings $x_1\ldots x_n$ that...
TAOCP 7.2.2.2 Exercise 3
Section 7.2.2.2: Satisfiability Exercise 3. ▶ [ M21 ] [M21] Exactly how many clauses are in waerden$(j, k; n)$? Verified: yes Solve time: 1m30s Solution By the definition of $\operatorname{waerden}(j,k;n)$, the clauses are divided into two families. The positive clauses are $$ (x_i\vee x_{i+d}\vee\cdots\vee x_{i+(j-1)d}) $$ with $$ 1\leq i\leq n-(j-1)d,\qquad d\geq1, $$ and the negative clauses are $$ (\bar{x} i\vee\bar{x} {i+d}\vee\cdots\vee\bar{x}_{i+(k-1)d}) $$ with $$ 1\leq i\leq n-(k-1)d,\qquad d\geq1. $$...
TAOCP 7.2.2.1 Exercise 97
Section 7.2.2.1: Dancing Links Exercise 97. [ M21 ] Find all $9 \times 9$ ternary orotoruses $D = {d_{ij}}$ that are symmetrical, in the sense that $d_{(i+3)\bmod 9} = (d_{ij}+1)\bmod 3$. (See exercise 7.2.1.1–109.) Verified: no Solve time: 3m37s The supplied statement is still insufficient to determine the mathematical answer. The previous response was right that the definition is essential, but it stopped too early. The missing information is not...
TAOCP 7.2.2.1 Exercise 79
Section 7.2.2.1: Dancing Links Exercise 79. [ M20 ] [M20] How many options are in (48), when they are completely listed? Verified: no Solve time: 2m55s Solution The statement of the exercise refers to equation (48), but equation (48) is not included in the supplied Section 7.2.2.1 excerpt. The number of options depends entirely on the exact definition of the option set in (48), so the answer cannot be derived...
TAOCP 7.2.2.1 Exercise 77
Section 7.2.2.1: Dancing Links Exercise 77. [ M21 ] [M21] Given graphs $G$ and $H$, each with $n$ vertices, use Algorithm X to decide whether or not $G$ is isomorphic to a subgraph of $H$. (In such a case we say that $G$ is embedded in H .) Verified: no Solve time: 3m05s Solution Let $$ G=(V,E),\qquad H=(W,F), $$ with $$ |V|=|W|=n. $$ We construct an exact cover instance whose...
TAOCP 7.2.2.1 Exercise 75
Section 7.2.2.1: Dancing Links Exercise 75. ▶ [ M24 ] [M24] A grope is a set $G$ together with a binary operation $\circ$, in which the identity $x \circ (y \circ x) = y$ is satisfied for all $x \in G$ and $y \in G$. a) Prove that the identity $(x \circ y) \circ x = y$ also holds, in every grope. b) Which of the following "multiplication tables" define...
TAOCP 7.2.2.1 Exercise 72
Section 7.2.2.1: Dancing Links Exercise 72. [ M23 ] [M23] Generate random instances of Dominosa, and estimate the probability of obtaining a $7 \times 8$ matrix with a unique solution. Use two models of randomness: (i) Each matrix whose elements are permutations of the multiset ${8 \times 0, 8 \times 1, \ldots, 8 \times 6}$ is equally likely; (ii) each matrix obtained from a random shuffle of the dominoes is...
TAOCP 7.2.2.1 Exercise 445
Section 7.2.2.1: Dancing Links Exercise 445. ▶ [ M22 ] It's surprisingly difficult to construct a valid hitori puzzle that has no seeds. In fact, there are no $n \times n$ examples for $n \le 9$ except when $n = 6$. But it turns out that quite a few seedless $6 \times 6$ hitori puzzles do exist. Consider the five hitori covers below. Determine, for each of them, the exact...
TAOCP 7.2.2.1 Exercise 442
Section 7.2.2.1: Dancing Links Exercise 442. ▶ [ M23 ] Enumerate all hitori covers of $P_m \square P_n$, for $1 \le m \le n \le 9$. Verified: no Solve time: 3m19s Working
TAOCP 7.2.2.1 Exercise 432
Section 7.2.2.1: Dancing Links Exercise 432. ▶ [ M25 ] We can't design new kakuro puzzles by randomly filling the blanks and using the resulting sums as the constraints, because the vast majority of possible sums yield nonunique solutions. Verify this experimentally for the generic diagrams In each case determine the exact number of ways to fill the blanks, without repeated digits in any row or column, as well as...
TAOCP 7.2.2.1 Exercise 423
Section 7.2.2.1: Dancing Links Exercise 423. ▶ [ M25 ] For each of the $(m-1)n + n(n-1)$ potential edges $e$ in the solution of an $m \times n$ masyu puzzle, let $x_e$ be the Boolean variable "$e$ is present". The XCC problem constructed in exercise 422 is essentially a set of constraints on those variables. Explain how to improve that construction dramatically, by exploiting the following special property that is...
TAOCP 7.2.2.1 Exercise 420
Section 7.2.2.1: Dancing Links Exercise 420. [ M22 ] Consider an $m \times n$ slitherlink with $m$ and $n$ odd, having 2s in the pattern $$ \begin{matrix} 2 & 2 & 2 & 2 & 2 & 2 \ 2 & 2 & 2 & 2 & 2 & 2 \ 2 & 2 & 2 & 2 & 2 & 2 \ \end{matrix} $$ (and possibly other clues). Show...
TAOCP 7.2.2.1 Exercise 42
Section 7.2.2.1: Dancing Links Exercise 42. [ M21 ] [M21] Explain how the method of exercise 40 could be extended to give representations of all solutions, instead of simply counting them. Verified: yes Solve time: 1m24s Solution The counting algorithm of exercise 40 loses information because each database entry $(s_j,c_j)$ stores only the number of ways to obtain the set $s_j$. To recover all solutions, replace the counter $c_j$ by...
TAOCP 7.2.2.1 Exercise 39
Section 7.2.2.1: Dancing Links Exercise 39. ▶ [ M21 ] $[M21]$ Experiment with exact cover problems that are defined by $m$ random options on $n$ items. (Each option is generated independently, with repetitions permitted.) a) Use a fixed probability $p$ that item $i$ is included in any given option. b) Let every option be a random sample of $r$ distinct items. Verified: no Solve time: 1m57s Solution Let $m$ be...
TAOCP 7.2.2.1 Exercise 38
Section 7.2.2.1: Dancing Links Exercise 38. [ M25 ] $[M25]$ Devise an efficient way to compute the sequence $\langle g_n \rangle$ of exercise 37. Verified: no Solve time: 2m21s Solution Let $g_n$ denote the lexicographically smallest solution of the $\infty$ queens problem. Thus $g_n$ is the row occupied by the queen in column $n$, and the defining condition is $$ g_i\ne g_j,\qquad |g_i-g_j|\ne |i-j| $$ for all distinct positive integers...
TAOCP 7.2.2.1 Exercise 376
Section 7.2.2.1: Dancing Links Exercise 376. ▶ [ M25 ] [M25] Find seven different rectangles of area $1/7$ that can be assembled into a square of area 1, and prove that the answer is unique. Verified: no Solve time: 5m48s $$ \textbf{Solution.} $$ It is convenient first to construct the dissection with a square of side $210$. Afterward, dividing all lengths by $210$ gives the required square of area $1$....
TAOCP 7.2.2.1 Exercise 368
Section 7.2.2.1: Dancing Links Exercise 368. [ M21 ] [M21] (H. Postl, 2017.) Show that an $m \times n$ motley dissection of order $t$ can exist only if $n < 2t/3$. Hint: Consider adjacent subrectangles. Verified: no Solve time: 3m07s Solution Let the $m\times n$ rectangle be divided into $t$ subrectangles. Number the subrectangles in a way that distinguishes adjacent ones only when they share a side. The defining property...
TAOCP 7.2.2.1 Exercise 361
Section 7.2.2.1: Dancing Links Exercise 361. [ M25 ] $[M25]$ The maximum number of subrectangles in a reduced $m \times n$ pattern is obviously $mn$. What is the minimum number? Verified: no Solve time: 6m24s Edit The minimum number of subrectangles in a reduced (m\times n) pattern is [ \boxed{m+n-1}. ] We first prove the necessary lower bound. Let a rectangular dissection consist of (r) rectangles. Let (a) be the...
TAOCP 7.2.2.1 Exercise 35
Section 7.2.2.1: Dancing Links Exercise 35. [ M21 ] $[M21]$ Continuing exercise 34, construct $A'$ having exactly three 1s in per column. Verified: no Solve time: 2m24s A mathematically correct solution cannot be written from the information provided because the exercise statement is incomplete. Exercise 35 is explicitly a continuation of Exercise 34: "Continuing exercise 34, construct $A'$ having exactly three 1s in every column." The matrix $A'$ requested in...
TAOCP 7.2.2.1 Exercise 347
Section 7.2.2.1: Dancing Links Exercise 347. ▶ [ M21 ] [M21] (N. G. de Bruijn, 1961.) Prove that an $l \times m \times n$ box can be completely filled with $1 \times 1 \times k$ bricks only if $k$ is a divisor of $l$, $m$, or $n$. (Consequently, it can be completely filled with $a \times b \times c$ bricks only if $a$, $b$, and $c$ all satisfy this condition.)...
TAOCP 7.2.2.1 Exercise 34
Section 7.2.2.1: Dancing Links Exercise 34. [ M25 ] $[M25]$ Given an exact cover problem $A$ as in exercise 33, construct an exact cover problem $A'$ such that (i) $A'$ has at most three 1s in every column; (ii) $A'$ and $A$ have exactly the same number of solutions. Verified: no Solve time: 6m27s $$ \textbf{Construction.} $$ Let $A$ be the given exact cover instance. We regard $A$ as a...
TAOCP 7.2.2.1 Exercise 33
Section 7.2.2.1: Dancing Links Exercise 33. [ M16 ] $[M16]$ Given an exact cover problem, specified by a $0$–$1$ matrix $A$, construct an exact cover problem $A'$ that has exactly one more solution than $A$ does. (Consequently it is NP-hard to determine whether an exact cover problem with at least one solution has more than one solution.) Assume that $A$ contains no all-zero rows. Verified: yes Solve time: 1m45s Solution...
TAOCP 7.2.2.1 Exercise 328
Section 7.2.2.1: Dancing Links Exercise 328. ▶ [ M23 ] [M23] Generalizing the first four examples of Fig. 75, study the set of all shapes obtainable by deleting three cubes from a $3 \times 3 \times 2$ box. (Two examples are shown here.) How many essentially different shapes are possible? Which shape is easiest? Which shape is hardest? Verified: no Solve time: 4m45s The statement of the exercise in the...
TAOCP 7.2.2.1 Exercise 326
Section 7.2.2.1: Dancing Links Exercise 326. ▶ [ M25 ] [M25] Use factorization to prove that Fig. 75's W-wall cannot be built. Verified: no Solve time: 2m25s Solution Assign coordinates $(x,y,z)$ to the cubies of Fig. 75's W-wall and color a cubie by the sign $$ \chi(x,y,z)=(-1)^{x+y+z}. $$ For any set of cubies $S$, define its parity polynomial by $$ F_S(X)=\sum_{(x,y,z)\in S}X^{x+y+z}. $$ A rigid motion of a Soma piece...
TAOCP 7.2.2.1 Exercise 323
Section 7.2.2.1: Dancing Links Exercise 323. [ M25 ] We've now seen polyominoes, polyiamonds, polyhexes, $\ldots$, polysticks, each of which have contributed new insights; and many other families of "polyforms" have in fact been studied. Let's close our survey with polykews , a relatively new family that seems worthy of further exploration. Polykews are the shapes that arise when we join squares alternately with rhombuses, in checkerboard fashion. For example,...
TAOCP 7.2.2.1 Exercise 31
Section 7.2.2.1: Dancing Links Exercise 31. [ M21 ] The running time of Algorithm X depends on the order of primary items in the active list, as well as on the order of options in the individual item lists. Explain how to randomize the algorithm so that (a) every item list is in random order after step X1; (b) step X3 chooses randomly among items with the minimum LB. Verified:...
TAOCP 7.2.2.1 Exercise 307
Section 7.2.2.1: Dancing Links Exercise 307. [ M21 ] If a $(3m+1) \times (3n+2)$ box is packed with $3mn+2m+n$ straight trominoes and one domino, where must the domino be placed? Verified: yes Solve time: 2m51s Solution Number the rows and columns of the rectangle starting with $0$. Thus the cells have coordinates $(r,c)$ with $0\le r\le 3m,\qquad 0\le c\le 3n+1.$ Color each cell by the residue of $r+c$ modulo $3$....
TAOCP 7.2.2.1 Exercise 304
Section 7.2.2.1: Dancing Links Exercise 304. [ M25 ] [M25] Prove that it's NP-complete to decide whether or not $n$ given polyominoes, each of which fits in a $6(\log n) \times 6(\log n)$ square, can be exactly packed into a square. Verified: no Solve time: 2m03s Solution Let $\mathcal P$ denote the decision problem in the statement. An instance of $\mathcal P$ consists of $n$ polyominoes, and the question is...
TAOCP 7.2.2.1 Exercise 28
Section 7.2.2.1: Dancing Links Exercise 28. [ M23 ] For what integers $c_0, t_0, c_1, t_1, \ldots, c_t, t_t$ with $1 \le c_j \le t_j$ does the text's formula (27) for estimated completion ratio give the value (a) $1/27$ (b) $1/37$? Verified: no Solve time: 1m31s Solution Formula (27) expresses the estimated completion ratio in the form $\prod_{j=0}^{t} \frac{c_j}{t_j}$ with integers satisfying $1 \le c_j \le t_j$. The requirement $\prod_{j=0}^{t}...
TAOCP 7.2.2.1 Exercise 264
Section 7.2.2.1: Dancing Links Exercise 264. [ M21 ] [M21] Suppose step Z3 always chooses the first active item $i = \texttt{RLINK}(0)$, instead of using the MIV heuristic, unless some other active item has $\text{LEN}(i) = 0$. Prove that Algorithm Z will then output an ordered ZDD. Verified: yes Solve time: 1m09s Solution Let the items be arranged in the circular doubly linked list headed by node $0$, with the...
TAOCP 7.2.2.1 Exercise 259
Section 7.2.2.1: Dancing Links Exercise 259. [ M25 ] Exercise 196 analyzes the behavior of Algorithm X on the bounded permutation problem defined by $a_1 \ldots a_n$. Show that Algorithm Z is considerably faster, by determining the number of options, ZDD nodes, and updates when $a_1 \ldots a_n$ is (a) $n \ldots n$ [with $n!$ solutions]; (b) $23 \ldots n$ [$n$ [with $2^{n-1}$ solutions]. Assume that the items are $X_1,...
TAOCP 7.2.2.1 Exercise 256
Section 7.2.2.1: Dancing Links Exercise 256. ▶ [ M23 ] What is the behavior of Algorithm Z when it is asked to find all perfect matchings of the "bizarre" graph (89)? Verified: no Solve time: 2m22s Solution Algorithm Z reduces the problem of finding perfect matchings of a graph to an exact cover instance in which each vertex is an item and each edge is an option covering its two...
TAOCP 7.2.2.1 Exercise 244
Section 7.2.2.1: Dancing Links Exercise 244. [ M21 ] [M21] The induced subgraphs $G \mid U$ of a graph or digraph $G$ have an interior cost , defined to be the number of ordered pairs of vertices in $U$ that are not adjacent. For example, the interior cost of option $(114)$ is 20, which is the maximum possible for six connected vertices of an undirected graph. Consider any exact cover...
TAOCP 7.2.2.1 Exercise 243
Section 7.2.2.1: Dancing Links Exercise 243. [ M20 ] [M20] Assume that every item $i$ of an XCC problem has been given a weight $w_i$, and that every solution to the problem involves exactly $d$ options. If the cost of every option is $$\binom{n}{k}^{-1}$, where $x_i$ is the value of the option's weights, prove that every minimum-cost solution also minimizes $\sum_{k=1}^{d}(x_k - r)^2$, for any given real number $r$. Verified:...
TAOCP 7.2.2.1 Exercise 242
Section 7.2.2.1: Dancing Links Exercise 242. ▶ [ M23 ] [M23] Extend the algorithm of exercise 7.2.2–78 so that it visits only subgraphs that don't cut off connected regions whose size isn't a sum of integers in $[L,.,.,U]$. Verified: no Solve time: 1m01s Solution Let $G = (V,E)$ be the graph processed by the algorithm of exercise 7.2.2–78, and let $H$ denote the current partial subgraph at any stage of...
TAOCP 7.2.2.1 Exercise 237
Section 7.2.2.1: Dancing Links Exercise 237. ▶ [ M21 ] True or false: Two solutions to the text's prime square problem cannot have the same product unless they are transposes of each other. Verified: no Solve time: 1m51s Solution Let a solution of the prime square problem be an $n \times n$ array $(x_{ij})$ of primes satisfying the defining constraints of the problem in the text, and let the product...
TAOCP 7.2.2.1 Exercise 236
Section 7.2.2.1: Dancing Links Exercise 236. ▶ [ M21 ] When the cost of a queen $(i,j)$ is $8d(i,j)^4$, for larger and larger values of $N$, the minimum-cost solutions to the $n$ queens problem eventually converge to a fixed pattern. And these "ultimate" solutions turn out to be quite attractive, indeed, this family of solutions is arguably the most beautiful of all! For example, the case $n = 16$, illustrated...
TAOCP 7.2.2.1 Exercise 234
Section 7.2.2.1: Dancing Links Exercise 234. [ M20 ] The minimum-cost placement of $n$ nonattacking queens when the cost of a queen in cell $(i,j)$ is $8d(i,j)^2$, the square of its distance from the center? Verified: no Solve time: 2m27s Solution Let the board be $n \times n$, and let the center be $\left(\frac{n+1}{2}, \frac{n+1}{2}\right).$ For a queen placed at $(i,j)$, the cost is $8d(i,j)^2,$ and in the standard geometric...
TAOCP 7.2.2.1 Exercise 224
Section 7.2.2.1: Dancing Links Exercise 224. ▶ [ M21 ] Construct an exact cover problem with $O(n)$ options that causes Algorithm P to perform $n$ rounds of reduction (that is, it executes step P2 $n$ times). Verified: yes Solve time: 1m51s Solution Let the items be $x_1, x_2, \dots, x_n$. Define an exact cover instance in which the set of options consists of the $n$ singleton options $$ O_i =...
TAOCP 7.2.2.1 Exercise 213
Section 7.2.2.1: Dancing Links Exercise 213. [ M21 ] [M21] If $\pi$ and $\pi'$ are two partitions of the same set, say that $\pi < \pi'$ if the restricted growth string of $\pi$ is lexicographically less than the restricted growth string of $\pi'$. Let $(\alpha, \beta; \alpha', \beta')$ be a canonical bipair in the sense of exercise 212. Also let $\pi$ be a partition of the items such that $\alpha$...
TAOCP 7.2.2.1 Exercise 212
Section 7.2.2.1: Dancing Links Exercise 212. ▶ [ M21 ] [M21] If the primary items of an exact cover problem have been linearly ordered, we can say that the bipair $(\alpha, \beta; \alpha', \beta')$ is canonical if (i) the smallest item in all four options appears in $\alpha$ and $\alpha'$; and (ii) option $\alpha$ is lexicographically smaller than option $\alpha'$, when their items have been listed in ascending order. a)...
TAOCP 7.2.2.1 Exercise 204
Section 7.2.2.1: Dancing Links Exercise 204. [ M25 ] [M25] True or false: Node $\alpha\alpha'$ is dominant in $T \oplus T'$ if and only if $\alpha$ is dominant in $T$ and $\alpha'$ is dominant in $T'$. Hint: Express $\deg(\alpha\alpha')$ in terms of $\deg(\alpha)$ and $\deg(\alpha')$. Verified: yes Solve time: 42s Solution Let $$ d=\deg(\alpha), \qquad d'=\deg(\alpha'). $$ By the definition of the operation $\oplus$ in the preceding section, the degree...
TAOCP 7.2.2.1 Exercise 199
Section 7.2.2.1: Dancing Links Exercise 199. [ M25 ] Analyze the behavior of Algorithm X on the 3D matching problem whose options are '$a_1 b_1 c_k$' for $1 \le i, j \le n$ and $1 \le k \le m$? $m = 1; n$). Verified: no Solve time: 4m58s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2...
TAOCP 7.2.2.1 Exercise 198
Section 7.2.2.1: Dancing Links Exercise 198. [ M25 ] Let $\pi_{rs} = \Pr{p_r = s}$, when $p$ is a random element of $P(a_1, \ldots, a_n)$. a) Compute these probabilities when $n = 9$ and $a_1 a_2 \ldots a_9 = 255667999$. b) If $r < r'$ and $s < s'$, show that $\pi_{rs}/\pi_{rs'} = \pi_{r's}/\pi_{r's'}$, when $\pi_{rs}\pi_{r's'} \ne 0$. Verified: no Solve time: 6m54s Setup We seek all integers $n <...
TAOCP 7.2.2.1 Exercise 197
Section 7.2.2.1: Dancing Links Exercise 197. [ M25 ] Let $P(a_1, \ldots, a_n)$ be the set of all permutations $p_1 \ldots p_n$ that solve the bounded permutation problem for $a_1 \ldots a_n$, given $a_1 \le a_2 \le \cdots \le a_n$ and $a_j \ge j$. a) Prove that $P(a_1, \ldots, a_n) \ge {(n!a_1) \ldots (2!a_2)(1!a_1) \mid j \le t_j \le a_n \text{ for } 1 \le j \le n}$. b) Also...
TAOCP 7.2.2.1 Exercise 195
Section 7.2.2.1: Dancing Links Exercise 195. ▶ [ M22 ] Approximately how many updates does Algorithm X perform, when it is asked to find all of the perfect matchings of the graph (8g)? Verified: no Solve time: 5m06s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying...
TAOCP 7.2.2.1 Exercise 188
Section 7.2.2.1: Dancing Links Exercise 188. [ M21 ] Prove that the $\langle \Xi_n \rangle = \langle 0, 1, 1, 3, 9, 1, 3, 9, 28, \ldots \rangle$ can be calculated rapidly by forming a triangle of numbers analogous to Peirce's triangle 7.2.1.5–(12): $$\begin{array}{ccccccccccc} & & & & & 0 & & & & & \ & & & & 1 & & 1 & & & & \ &...
TAOCP 7.2.2.1 Exercise 186
Section 7.2.2.1: Dancing Links Exercise 186. [ M24 ] Consider $t = 0$ when Algorithm X is given the extreme problem of order $n$. a) How many updates, $u_n$, does it perform when covering $i$ in step X4? b) How many does it perform in step X5, when the option containing $x_0$ has size $k$? c) Therefore derive (84). Verified: no Solve time: 4m16s Setup We seek all integers $n...
TAOCP 7.2.2.1 Exercise 185
Section 7.2.2.1: Dancing Links Exercise 185. [ M22 ] What is the largest number of solutions to a strict exact cover problem that has $N_1$ primary items and $N_2$ secondary items? Verified: no Solve time: 1m31s Solution A strict exact cover problem consists of options, each option containing exactly one primary item and any number of secondary items, such that every primary item is covered exactly once and each secondary...
TAOCP 7.2.2.1 Exercise 184
Section 7.2.2.1: Dancing Links Exercise 184. ▶ [ M22 ] Suppose $0 \le t \le 2t_{\infty}$. Is there a strict exact cover problem with $n$ items that has exactly $t$ solutions? (For example, consider the case $n = 9$, $t = 10000$.) Verified: no Solve time: 4m51s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots...
TAOCP 7.2.2.1 Exercise 181
Section 7.2.2.1: Dancing Links Exercise 181. [ M20 ] Find constants $c_r$ such that $\tilde{D}(5n + r) = 4^n c_r - \frac{3}{4}$ for $n \ge 2$ and $0 \le r < 5$. Verified: no Solve time: 3m14s Solution Assume $$ \tilde{D}(5n+r)=4^n c_r-\frac{3}{4}, \qquad n\ge 2,\quad 0\le r<5. $$ Fix $r$ and substitute $n$ and $n+1$: $$ \tilde{D}(5(n+1)+r)=4^{n+1}c_r-\frac{3}{4}, $$ $$ \tilde{D}(5n+r)=4^n c_r-\frac{3}{4}. $$ Eliminating $c_r$ between these two expressions gives $$...
TAOCP 7.2.2.1 Exercise 178
Section 7.2.2.1: Dancing Links Exercise 178. [ M23 ] [M23] [ Factorization of an integer. ] Use Algorithm M to find all representations of 360 as a product $n_1 \cdot n_2 \cdots n_k$, where (a) $1 < n_1 \le \cdots < n_k$, (b) $2 \le n_1 \le \cdots \le n_k$, Verified: no Solve time: 4m58s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2...
TAOCP 7.2.2.1 Exercise 177
Section 7.2.2.1: Dancing Links Exercise 177. [ M21 ] [M21] Algorithm 7.2.1.5M generates the $p(n_1, \ldots, n_m)$ partitions of the multiset ${1^{n_1} 2^{n_2} \cdots m^{n_m}}$ into submultisets. Consider the special cases where $n_1 = \cdots = n_k = 1$ and $n_{k+1} = \cdots = n_m = 2$ and $s + l = m$. a) Generate those partitions with Algorithm M, using the previous two exercises. b) Also generate the $q(n_1,...
TAOCP 7.2.2.1 Exercise 175
Section 7.2.2.1: Dancing Links Exercise 175. ▶ [ M21 ] [M21] Given an $M \times N$ matrix $A = (a_{ij})$ of 0s and 1s, explain how to find all vectors $x = (x_1 \ldots x_M)$ with $0 \le x_i \le a_i$ for $1 \le i \le M$ such that $xA = (y_1 \ldots y_N)$, where $a_i \le y_j \le v_j$ for $1 \le j \le N$. (This generalizes the MCC...
TAOCP 7.2.2.1 Exercise 149
Section 7.2.2.1: Dancing Links Exercise 149. [ M22 ] (Vertex-colored tetrahedra.) The graph $\text{simplex}(3,3,3,3,3,0)$ is a tetrahedron of side 3 with 20 vertices. It has 60 edges, which come from 10 unit tetrahedra. There are ten ways to color the vertices of a unit tetrahedron with four of the five colors ${\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}, \mathbf{e}}$, because mirror reflections are distinct. Can those ten colored tetrahedra be packed into $\text{simplex}(3,3,3,3,3,0)$,...
TAOCP 7.2.2.1 Exercise 143
Section 7.2.2.1: Dancing Links Exercise 143. ▶ [ M25 ] $[M25]$ The graph $simplex(n, a, b, c, 0, 0, 0)$ in the Stanford GraphBase is the truncated triangular grid consisting of all vertices $xyz$ such that $$x + y + z = n,\quad 0 \le x \le a,\quad 0 \le y \le b,\quad 0 \le z \le c.$$ Two vertices are adjacent if their coordinates all differ by at most...
TAOCP 7.2.2.1 Exercise 125
Section 7.2.2.1: Dancing Links Exercise 125. [ M20 ] When a set of $s$ triangles is magnified by an integer $k$, we obtain $sk^2$ triangles. Describe the coordinates of those triangles, in terms of the coordinates of the originals, using the system of exercise 124. Verified: no Solve time: 5m15s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n =...
TAOCP 7.2.2.1 Exercise 124
Section 7.2.2.1: Dancing Links Exercise 124. [ M22 ] Devise a system of coordinates for representing the positions of equilateral triangles in patterns such as (59). Represent also the edges between them. Verified: no Solve time: 5m28s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1...
TAOCP 7.2.2.1 Exercise 116
Section 7.2.2.1: Dancing Links Exercise 116. ▶ [ M25 ] Given a graph $G$ on vertices $V$, let $\mu(G)$ be obtained by (i) adding new vertices $V' = {v' \mid v \in V}$, with $u' \mathbin{-!!-} v$ when $u \mathbin{-!!-} v$; and also (ii) adding another vertex $w$, with $w \mathbin{-!!-} v'$ for all $v' \in V'$. If $G$ has $m$ edges and $n$ vertices, $\mu(G)$ has $3m+n$ edges and...
TAOCP 7.2.2.1 Exercise 115
Section 7.2.2.1: Dancing Links Exercise 115. [ M25 ] Continuing exercise 114, how many hypersudoku solutions have automorphisms of the following types? (a) transposition; (b) the transformation of exercise 67(d); (c) 90° rotation; (d) both (b) and (c). Verified: no Solve time: 5m07s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one...
TAOCP 7.2.2.1 Exercise 114
Section 7.2.2.1: Dancing Links Exercise 114. [ M25 ] Let $\alpha$ be a permutation of the cells of a $9 \times 9$ array that takes any sudoku solution into another sudoku solution. We say that $\alpha$ is an automorphism of the sudoku solution $S = (s_{ij})$ if there's a permutation $\pi$ of ${1, 2, \ldots, 9}$ such that $s_{ij\alpha} = \pi s_{ij}$ for $0 \le i, j \le 9$. For...
TAOCP 7.2.2.1 Exercise 1
Section 7.2.2.1: Dancing Links Exercise 1. ▶ [ M25 ] A doubly linked list of $n$ elements, with a list head at 0, begins with $\text{LLINK}(k) = k - 1$ and $\text{RLINK}(k - 1) = k$ for $1 \le k \le n$; furthermore $\text{LLINK}(0) = n$ and $\text{RLINK}(n) = 0$, as in (3). But after we use operation (1) to delete elements $a_1, a_2, \ldots, a_n$, where $a_1 a_2 \ldots...
TAOCP 7.2.2 Exercise 77
Section 7.2.2: Backtracking Exercise 77. [ M22 ] A $v$-reachable subset of a directed graph $G$ is a nonempty set of vertices $H$ with the property that every $u \in H$ can be reached from $v$ by at least one oriented path in $G\langle H\rangle$. (In particular, $v$ itself must be in $H$.) a) The digraph $P_3' \mathbin{\square} P_3'$ is like $P_3 \mathbin{\square} P_3$ except that all arcs between vertices...
TAOCP 7.2.1.6 Exercise 88
Section 7.2.1.6: Generating All Trees Exercise 88. [ M20 ] Analyze Algorithm O: How often is step O4 performed? What is the total number of times $p_k$ is changed in step O5? Verified: no Solve time: 3m53s The previous solution failed by tying the execution of step O4 to a “parent-to-child transition” interpretation rather than to the actual control structure of Algorithm O. The correct way to reason is to...
TAOCP 7.2.1.6 Exercise 83
Section 7.2.1.6: Generating All Trees Exercise 83. [ M20 ] (G. Hansel.) Show that there are at most $3^{M_n}$ monotone Boolean functions $f(x_1, \ldots, x_n)$ of $n$ Boolean variables. Verified: no Solve time: 5m44s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge...
TAOCP 7.2.1.6 Exercise 71
Section 7.2.1.6: Generating All Trees Exercise 71. [ M21 ] [M21] Let $M_{\sigma s}$ be the size of the largest possible set $S$ of $n$-bit strings with the property that, if $\sigma$ and $\tau$ are members of $S$ with $\sigma \le \tau$, then $\nu(\sigma) < \nu(\tau) + t$. (Thus, for example, $M_{n0} = M_n$ by Sperner's theorem.) Find a formula for $M_{ns}$. Verified: no Solve time: 2m14s Solution Let $B_n...
TAOCP 7.2.1.6 Exercise 67
Section 7.2.1.6: Generating All Trees Exercise 67. [ M22 ] [M22] What is the generating function $S(z) = \sum_n S_n z^n$ for Schröder numbers? Verified: no Solve time: 3m10s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$...
TAOCP 7.2.2 Exercise 56
Section 7.2.2: Backtracking Exercise 56. ▶ [ M25 ] The commafree code search procedure in Algorithm C doesn't actually fit the mold of Algorithm E, because it incorporates lookahead, dynamic ordering, reversible memory, and other enhancements to the basic backtrack paradigms. How could its running time be reliably estimated with Monte Carlo methods? Verified: no Solve time: 5m07s Setup We seek all integers $n < 10^9$ such that the equation...
TAOCP 7.2.2 Exercise 54
Section 7.2.2: Backtracking Exercise 54. [ M21 ] Instead of choosing each $y_i$ in step E5 with probability $1/d$, we could use a biased distribution where $\Pr{I = i \mid X_1, \ldots, X_{l-1}} = p_{X_1 \cdots X_{l-1}}(i) > 0$. How should the estimate $S$ be modified so that its expected value in this general scheme is still $C(t)$? Verified: no Solve time: 4m15s Setup We seek all integers $n <...
TAOCP 7.2.2 Exercise 51
Section 7.2.2: Backtracking Exercise 51. [ M22 ] $[M22]$ What's a simple martingale that corresponds to Theorem E? Verified: no Solve time: 5m07s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$ be the number of terms among...
TAOCP 7.2.1.6 Exercise 56
Section 7.2.1.6: Generating All Trees Exercise 56. [ M25 ] $[M25]$ (Rényi–Szpankowski, 1990.) Prove the summation formula $$\sum_{k=0}^{m} C_k C_{n-1-k} = \frac{2n-m}{2n(n+1)} \binom{2n}{m} \binom{2n-2m}{n-m}, \quad \text{for } 0 \le m \le n.$$ Verified: no Solve time: 6m42s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1...
TAOCP 7.2.1.6 Exercise 52
Section 7.2.1.6: Generating All Trees Exercise 52. [ M23 ] $[M23]$ Find the mean and variance of the quantity $d_n$ in Table 1, when nested parentheses $a_1 \ldots a_{2n}$ are chosen at random. Verified: no Solve time: 6m02s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying...
TAOCP 7.2.1.6 Exercise 51
Section 7.2.1.6: Generating All Trees Exercise 51. [ M23 ] $[M23]$ Let $\hat{z}_1 \ldots \hat{z}_n$ be the $N$th combination of ${1, 2, \ldots, 2n}$ with respect to $2n$ in another words, $\hat{z}_j = 2n - z_j$, where $z_j$ is defined in (8). Show that if $\hat{z}_1 \hat{z}_2 \ldots \hat{z}_n$ is the $(N+1)$st $n$-combination of ${0, 1, \ldots, 2n-1}$ generated by Algorithm 7.2.1.3L, then $z_1 z_2 \ldots z_n$ is the $(N...
TAOCP 7.2.1.6 Exercise 42
Section 7.2.1.6: Generating All Trees Exercise 42. [ M22 ] How many unlabeled forests with $n$ nodes are (a) self-conjugate? (b) self-transpose? (c) self-dual? (See exercises 11, 12, 19, and 26.) Verified: no Solve time: 6m09s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge...
TAOCP 7.2.1.6 Exercise 41
Section 7.2.1.6: Generating All Trees Exercise 41. [ M21 ] Show that the ballot numbers have a simple generating function $\sum C_{pq} n^p z^q$. Verified: no Solve time: 6m02s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$...
TAOCP 7.2.1.6 Exercise 40
Section 7.2.1.6: Generating All Trees Exercise 40. [ M25 ] (a) Prove that $C_{pq}$ is odd if and only if $p\ &\ (q+1) = 0$, in the sense that the binary representations of $p$ and $q+1$ have no bits in common. (b) Therefore $C_n$ is odd if and only if $n+1$ is a power of 2. Verified: no Solve time: 6m07s Setup We seek all integers $n < 10^9$ such...
TAOCP 7.2.2 Exercise 21
Section 7.2.2: Backtracking Exercise 21. ▶ [ M25 ] [M25] If $x = x_1 x_2 \ldots x_{2n}$, let $x^D = (-x_{2n}) \ldots (-x_2)(-x_1) = -x^R$ be its dual. a) Show that if $x$ is odd and $x$ solves Langford's problem $(n)$, we have $x_k = n$ for some $k \le \lfloor n/2 \rfloor$ if and only if $x_k^D = n$ for some $k \le \lfloor n/2 \rfloor$. b) Find a...
TAOCP 7.2.1.6 Exercise 38
Section 7.2.1.6: Generating All Trees Exercise 38. [ M22 ] What is the total number of memory references performed by Algorithm L, as a function of $n$? Verified: no Solve time: 5m59s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge...
TAOCP 7.2.1.6 Exercise 36
Section 7.2.1.6: Generating All Trees Exercise 36. ▶ [ M25 ] Analyze the ternary tree generation algorithm of exercise 20(b). Hint: There are $(2n+1)^{-1}\binom{3n}{n}$ ternary trees with $n$ internal nodes, by exercise 2.3.4.4–11. Verified: no Solve time: 6m03s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying...
TAOCP 7.2.1.6 Exercise 34
Section 7.2.1.6: Generating All Trees Exercise 34. [ M25 ] (R. P. Stanley.) Show that the number of maximal chains in the Stanley lattice of order $n$ is $(n(n-1)/2)!/(1^{n-1}3^{n-2}\cdots(2n-3)^1(2n-3)!)$. Verified: no Solve time: 5m50s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge...
TAOCP 7.2.2 Exercise 14
Section 7.2.2: Backtracking Exercise 14. [ M25 ] [M25] If exercise 12 has $T(n)$ toroidal solutions, show that $Q(mn) \ge Q(m)^2 T(n)$. Verified: no Solve time: 5m03s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$ be the...
TAOCP 7.2.2 Exercise 11
Section 7.2.2: Backtracking Exercise 11. [ M25 ] [M25] (W. Ahrens, 1910.) Both solutions of the $n$ queens problem when $n = 4$ have quartersurn symmetry : Rotation by 90° leaves them unchanged, but reflection doesn't. a) Can the $n$ queens problem have a solution with reflection symmetry? b) Show that quarterturn symmetry is impossible if $n \bmod 4 \in {2, 3}$. c) Sometimes the solution to an $n$ queens...
TAOCP 7.2.1.6 Exercise 17
Section 7.2.1.6: Generating All Trees Exercise 17. [ M16 ] [M16] Characterize all unlabeled forests $F$ such that $F^{BT} = F^{TR}$. (See exercise 14.) Verified: no Solve time: 5m45s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$...
TAOCP 7.2.1.6 Exercise 107
Section 7.2.1.6: Generating All Trees Exercise 107. [ M24 ] Determine the aspects of all connected graphs that have $n \le 5$ vertices and no self-loops or parallel edges. Verified: no Solve time: 5m28s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge...
TAOCP 7.2.1.6 Exercise 10
Section 7.2.1.6: Generating All Trees Exercise 10. [ M20 ] [M20] ( Worm walks. ) Given a string of nested parentheses $a_1 a_2 \ldots a_{2n}$, let $w_j$ be the excess of left parentheses over right parentheses in $a_1 a_2 \ldots a_j$, for $0 \le j \le 2n$. Prove that $w_0 + w_1 + \cdots + w_{2n} = 2(c_1 + \cdots + c_n) + n$. Verified: no Solve time: 5m49s Setup...
TAOCP 7.2.1.5 Exercise 80
Section 7.2.1.5: Generating All Set Partitions Exercise 80. [ M25 ] Prove that universal sequences for ${1, 2, \ldots, n}$ exist in the sense of the previous exercise whenever $n \ge 4$. Verified: no Solve time: 5m48s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1...
TAOCP 7.2.1.5 Exercise 71
Section 7.2.1.5: Generating All Set Partitions Exercise 71. [ M20 ] How many partitions of ${n_1, \ldots, n_m, m}$ have exactly 2 parts? Verified: no Solve time: 5m52s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$ be...
TAOCP 7.2.1.5 Exercise 37
Section 7.2.1.5: Generating All Set Partitions Exercise 37. [ M18 ] Alexander Pushkin adopted an elaborate structure in his poetic novel Eugene Onegin (1833), based not only on "masculine" rhymes in which the sounds of accented final syllables agree with each other (pain–rain, form–warm, pun–fun, bucks–crux), but also on "feminine" rhymes in which one or two unstressed syllables also participate (humor–tumor, tetrameter–pentameter, lecture–conjecture, iguana–piranha). Every stanza of Eugene Onegin is...
TAOCP 7.2.1.5 Exercise 36
Section 7.2.1.5: Generating All Set Partitions Exercise 36. [ M21 ] [M21] Continuing exercise 35, what is the generating function $\sum_n \varpi'_n z^n/n!$? Verified: no Solve time: 5m42s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$ be...
TAOCP 7.2.1.5 Exercise 35
Section 7.2.1.5: Generating All Set Partitions Exercise 35. [ M22 ] [M22] Let $\varpi'_n$ be the number of $n$-line poems that are "completely rhymed," in the sense that every line rhymes with at least one other. Then we have $(\varpi'_0, \varpi'_1, \varpi'_2, \ldots) = (1, 0, 1, 1, 4, 11, 41, \ldots)$. Give a combinatorial proof of the fact that $\varpi' n = \varpi' {n+1} - \varpi_n$. Verified: no Solve...
TAOCP 7.2.1.5 Exercise 33
Section 7.2.1.5: Generating All Set Partitions Exercise 33. [ M21 ] [M21] How many partitions of ${1, 2, \ldots, n}$ are there in which every block has at most $k-1$ elements, where $n \bmod 6 = (1, 2, 3, 4, 5, 0)$? Prove that $\delta_n = (-1, 0, -1, 0, 1, 0)$ when $n \bmod 6 = (1, 2, 3, 4, 5, 0)$. Verified: no Solve time: 5m41s Setup We...
TAOCP 7.2.1.5 Exercise 32
Section 7.2.1.5: Generating All Set Partitions Exercise 32. [ M22 ] [M22] Let $\delta_n$ be the number of restricted growth strings $a_1 \ldots a_n$ for which the sum $a_1 + \cdots + a_n$ is even minus the number for which $a_1 + \cdots + a_n$ is odd. Prove that $$\delta_n = (-1,, 0,, -1,, 0,, 1,, 0) \quad \text{when } n \bmod 6 = (1,, 2,, 3,, 4,, 5,, 0).$$...
TAOCP 7.2.1.5 Exercise 28
Section 7.2.1.5: Generating All Set Partitions Exercise 28. ▶ [ M25 ] [M25] ( Generalized rook polynomials. ) Consider an arrangement of $a_1 + a_2 + \cdots + a_k$ square cells in rows and columns, where row $k$ contains cells in columns $1, \ldots, a_k$. Place zero or more "rooks" into the cells, with at most one rook in each row and at most one in each column. An empty...
TAOCP 7.2.1.5 Exercise 15
Section 7.2.1.5: Generating All Set Partitions Exercise 15. ▶ [ M21 ] What is the final partition generated by the algorithm of exercise 14? Verified: no Solve time: 5m45s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let $k$...
TAOCP 7.2.1.5 Exercise 9
Section 7.2.1.5: Generating All Set Partitions Exercise 9. [ M20 ] [M20] How many restricted growth strings $a_1 \ldots a_n$ contain exactly $k_j$ occurrences of $j$, given the integers $k_0, k_1, \ldots, k_{n-1}$? Verified: no Solve time: 9m31s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying...
TAOCP 7.2.1.5 Exercise 7
Section 7.2.1.5: Generating All Set Partitions Exercise 7. [ M20 ] [M20] How many permutations $a_1 \ldots a_n$ of ${1, \ldots, n}$ have the property that $a_{k-1} > a_k > a_j$ implies $j > k$? Verified: no Solve time: 5m41s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive...
TAOCP 7.2.1.5 Exercise 3
Section 7.2.1.5: Generating All Set Partitions Exercise 3. [ M23 ] [M23] What is the millionth partition of ${1, \ldots, 12}$ generated by Algorithm H? Verified: no Solve time: 5m42s Setup We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$. Let...
TAOCP 7.2.1.4 Exercise 73
Section 7.2.1.4: Generating All Partitions Exercise 73. [ M25 ] [M25] Suppose we write down all partitions of n, for example 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111 when n = 6, and change each jth occurrence of k to j in each one: 1, 11, 11, 112, 12, 111, 1123, 123, 1212, 11234, 123456. a) Prove that this operation yields a permutation of the individual...
TAOCP 7.2.1.4 Exercise 68
Section 7.2.1.4: Generating All Partitions Exercise 68. [ M23 ] [M23] What partition of n into m parts has the largest product a1 . . . am, when (a) m is given; (b) m is arbitrary? Verified: no Solve time: 5m46s Solution Let the perfect partition be a multiset with distinct values $v_1 < v_2 < \cdots < v_t$, where each value $v_i$ occurs with multiplicity $b_i-1 \ge 0$. A...
TAOCP 7.2.1.4 Exercise 67
Section 7.2.1.4: Generating All Partitions Exercise 67. [ M25 ] [M25] (P. A. MacMahon, 1886.) A perfect partition of n is a multiset that has exactly n+1 submultisets, and these multisets are partitions of the integers 0, 1, . . . , n. For example, the multisets {1,1,1,1,1}, {2,2,1}, and {3,1,1} are perfect partitions of 5. Explain how to construct the perfect partitions of n that have fewest elements. Verified:...
TAOCP 7.2.1.4 Exercise 66
Section 7.2.1.4: Generating All Partitions Exercise 66. ▶ [ M25 ] [M25] (P-partitions.) Instead of insisting that a1 ≥a2 ≥· · · , suppose we want to consider all nonnegative compositions of n that satisfy a given partial order. For example, P. A. MacMahon observed that all solutions to the “up-down” inequalities a4 ≤a2 ≥a3 ≤a1 can be divided into five nonoverlapping types: a1 ≥a2 ≥a3 ≥a4; a1 ≥a2 ≥a4...
TAOCP 7.2.1.4 Exercise 59
Section 7.2.1.4: Generating All Partitions Exercise 59. [ M22 ] [M22] The Gray path (59) is symmetrical in the sense that the reversed sequence 6, 51, . . . , 111111 is the same as the conjugate sequence (111111)T, (21111)T, . . . , (6)T. Find all Gray paths α1, . . . , αp(n) that are symmetrical in this way. Verified: no Solve time: 44m15s Correctness The solution does...
TAOCP 7.2.1.4 Exercise 58
Section 7.2.1.4: Generating All Partitions Exercise 58. [ M23 ] [M23] (Symmetrical means.) Let α = a1 . . . am and β = b1 . . . bm be partitions of n. Prove that the inequality 1 m! xa1 p1 . . . xam pm ≥ 1 m! xb1 p1 . . . xbm pm holds for all nonnegative values of the variables (x1, . . ....
TAOCP 7.2.1.4 Exercise 57
Section 7.2.1.4: Generating All Partitions Exercise 57. [ M22 ] [M22] Suppose a matrix (aij) of 0s and 1s has row sums ri = j aij and column sums cj = i aij. By permuting rows and columns we can assume that r1 ≥r2 ≥· · · and c1 ≥c2 ≥· · · . Then λ = r1r2 . . . and µ = c1c2 . . ....
TAOCP 7.2.1.4 Exercise 53
Section 7.2.1.4: Generating All Partitions Exercise 53. ▶ [ M21 ] [M21] What is the millionth partition generated by Algorithm H when m = 32 and n = 100? Hint: 999999 = 80 12 + 66 11 + 50 7 + 41 6 + 33 5 + 26 4 + 21 4 . Verified: no Solve time: 15m27s Solution Let $a_1 \ge a_2 \ge \cdots \ge a_{32} \ge 1$ with...
TAOCP 7.2.1.4 Exercise 52
Section 7.2.1.4: Generating All Partitions Exercise 52. ▶ [ M21 ] [M21] What is the millionth partition generated by Algorithm P when n = 64? Hint: p(64) = 1741630 = 1000000 + 77 13 + 60 10 + 47 8 + 35 5 + 27 3 + 22 2 + 18 1 + 15 0 . Verified: no Solve time: 4m20s The statement of Exercise 7.2.1.4.50 is incomplete in the...
TAOCP 7.2.1.4 Exercise 46
Section 7.2.1.4: Generating All Partitions Exercise 46. [ M20 ] [M20] In the text’s analysis of Algorithm P, which is larger, T ′ 2(n) or T ′′ 2 (n)? Verified: no Solve time: 8m36s Solution Let $S$ be a multiset of positive integers, and write its distinct values in increasing order as $$ 1 \le b_1 < b_2 < \cdots < b_t, $$ with multiplicities $m_1, m_2, \ldots, m_t$. A...
TAOCP 7.2.1.4 Exercise 44
Section 7.2.1.4: Generating All Partitions Exercise 44. ▶ [ M22 ] [M22] How many partitions of n have their two smallest parts equal? Verified: no Solve time: 8m37s Solution Let $f_{l,m}(n)$ denote the number of partitions of $n$ having exactly $m$ parts and largest part equal to $l$. Let the ordinary generating function be $$ F_{l,m}(x) = \sum_{n \ge 0} f_{l,m}(n)x^n. $$ Every such partition can be written uniquely in...
TAOCP 7.2.1.4 Exercise 43
Section 7.2.1.4: Generating All Partitions Exercise 43. [ M18 ] [M18] Given n and k, how many partitions of n have a1 > a2 > · · · > ak? Verified: no Solve time: 8m56s Solution Let $f_{l,m}(n)$ denote the number of partitions of $n$ having exactly $m$ parts and largest part equal to $l$. Let the ordinary generating function be $$ F_{l,m}(x) = \sum_{n \ge 0} f_{l,m}(n)x^n. $$ Every...
TAOCP 7.2.1.4 Exercise 40
Section 7.2.1.4: Generating All Partitions Exercise 40. ▶ [ M25 ] [M25] (F. Franklin.) Generalizing Theorem C, show that, for 0 ≤k ≤m, [zn] (1 −zl+1) . . . (1 −zl+k) (1 −z)(1 −z2) . . . (1 −zm) is the number of partitions a1a2 . . . of n into m or fewer parts with the property that a1 ≤ak+1 + l. Verified: no Solve time: 8m36s Solution Let...
TAOCP 7.2.1.4 Exercise 39
Section 7.2.1.4: Generating All Partitions Exercise 39. [ M20 ] [M20] (A. Cauchy.) Continuing exercise 38, what is the generating function for the number of partitions into m parts, all distinct and less than l? Verified: no Solve time: 8m18s Solution Let $f_{l,m}(n)$ denote the number of partitions of $n$ having exactly $m$ parts and largest part equal to $l$. Let the ordinary generating function be $$ F_{l,m}(x) = \sum_{n...
TAOCP 7.2.1.4 Exercise 38
Section 7.2.1.4: Generating All Partitions Exercise 38. [ M20 ] [M20] Given positive integers l and m, what generating function enumerates partitions that have exactly m parts, and largest part l? (See Eq. (51).) Verified: no Solve time: 19m15s Solution Let $f_{l,m}(n)$ denote the number of partitions of $n$ having exactly $m$ parts and largest part equal to $l$. Let the ordinary generating function be $$ F_{l,m}(x) = \sum_{n \ge...
TAOCP 7.2.1.4 Exercise 37
Section 7.2.1.4: Generating All Partitions Exercise 37. [ M22 ] [M22] Prove the inclusion-exclusion bracketing lemma (48), by analyzing how many times a partition that has exactly q different parts exceeding m is counted in the rth partial sum. Verified: no Solve time: 8m36s Setup Let $\lambda = (\lambda_1 \ge \lambda_2 \ge \cdots)$ and $\mu = (\mu_1 \ge \mu_2 \ge \cdots)$ be partitions of the same integer $n$ such that...
TAOCP 7.2.1.4 Exercise 31
Section 7.2.1.4: Generating All Partitions Exercise 31. [ M24 ] [M24] (A. De Morgan, 1843.) Show that n 2 = ⌊n/2⌋and n 3 = ⌊(n2 + 6)/12⌋; find a similar formula for n 4 . Verified: no Solve time: 26m02s Solution Let $\left| \begin{matrix} n \ k \end{matrix} \right|$ denote the number of partitions of $n$ into exactly $k$ parts, equivalently the number of partitions of $n$ whose Ferrers diagram...
TAOCP 7.2.1.4 Exercise 30
Section 7.2.1.4: Generating All Partitions Exercise 30. [ M17 ] [M17] Find closed forms for the sums (a) k≥0 n −km m −1 and (b) k≥0 n m −k (which are finite, because the terms being summed are zero when k is large). Verified: no Solve time: 14m41s Solution Let $m \ge 1$ and $n \ge 0$. Throughout, the binomial coefficient $\left|\begin{matrix} a \ b...
TAOCP 7.2.1.4 Exercise 29
Section 7.2.1.4: Generating All Partitions Exercise 29. ▶ [ M16 ] [M16] Generalizing (41), evaluate the sum a1≥a2≥···≥am≥1 za1 1 za2 2 . . . zam m . Verified: no Solve time: 5m30s Setup Let $A_k(n)$ denote the Hardy–Ramanujan–Rademacher coefficient defined in equation (34) of Section 7.2.1.4. In Lehmer’s formulation, $A_k(n)$ is a finite exponential sum depending only on $k$ and the quadratic discriminant $D(n)=1-24n,$ with multiplicative dependence on...
TAOCP 7.2.1.4 Exercise 21
Section 7.2.1.4: Generating All Partitions Exercise 21. [ M21 ] [M21] (L. Euler.) Let q(n) be the number of partitions of n into distinct parts. What is a good way to compute q(n) if you already know the values of p(1), . . . , p(n)? Verified: no Solve time: 8m18s Solution Let $$ F(a,b;u,v)=\sum_{k,l\ge 0} u^k v^l z^{kl} \frac{(z-az)(z-az^2)\cdots(z-az^k)}{(1-z)(1-z^2)\cdots(1-z^k)} \frac{(z-bz)(z-bz^2)\cdots(z-bz^l)}{(1-z)(1-z^2)\cdots(1-z^l)}. $$ Rewrite each finite product in $q$-shifted factorial form....
TAOCP 7.2.1.4 Exercise 20
Section 7.2.1.4: Generating All Partitions Exercise 20. ▶ [ M21 ] [M21] Approximately how long does it take to compute a table of the partition numbers p(n) for 1 ≤n ≤N, using Euler’s recurrence (20)? Verified: no Solve time: 7m17s Solution Let $$ F(a,b;u,v)=\sum_{k,l\ge 0} u^k v^l z^{kl} \frac{(z-az)(z-az^2)\cdots(z-az^k)}{(1-z)(1-z^2)\cdots(1-z^k)} \frac{(z-bz)(z-bz^2)\cdots(z-bz^l)}{(1-z)(1-z^2)\cdots(1-z^l)}. $$ Rewrite each finite product in $q$-shifted factorial form. For $k\ge 0$, $$ (z-az^i)=z(1-az^{i-1}),\quad 1\le i\le k, $$ so $$...
TAOCP 7.2.1.4 Exercise 19
Section 7.2.1.4: Generating All Partitions Exercise 19. [ M22 ] [M22] (E. Heine, 1847.) Prove the four-parameter identity ∞ m=1 (1−wxzm)(1−wyzm) (1−wzm)(1−wxyzm) = ∞ k=0 wk(x−1)(x−z) . . . (x−zk−1)(y−1)(y−z) . . . (y−zk−1)zk (1−z)(1−z2) . . . (1−zk)(1−wz)(1−wz2) . . . (1−wzk) . Hint: Carry out the sum over either k or l in the formula k,l≥0 ukvlzkl (z −az)(z −az2) . . . (z −azk)...
TAOCP 7.2.1.4 Exercise 18
Section 7.2.1.4: Generating All Partitions Exercise 18. ▶ [ M23 ] [M23] (Doron Zeilberger.) Show that there is a one-to-one correspondence be- tween pairs of integer sequences (a1, a2, . . . , ar; b1, b2, . . . , bs) such that a1 ≥a2 ≥· · · ≥ar, b1 > b2 > · · · > bs, and pairs of integer sequences (c1, c2, . . . , cr+s;...
TAOCP 7.2.1.4 Exercise 16
Section 7.2.1.4: Generating All Partitions Exercise 16. [ M21 ] [M21] Find a formula for m,n p(k, m, n)wmzn, where p(k, m, n) is the number of partitions of n that have m parts and trace k. Sum it on k to obtain a nontrivial identity. 7.2.1.4 GENERATING ALL PARTITIONS 409 Verified: no Solve time: 6m39s Solution A partition of $n$ has trace $k$ when its Ferrers diagram has...
TAOCP 7.2.1.4 Exercise 15
Section 7.2.1.4: Generating All Partitions Exercise 15. [ M20 ] [M20] (J. J. Sylvester.) Find a generating function for the number of partitions that are self-conjugate (namely, partitions such that α = αT ). Verified: no Solve time: 24m20s Solution Let $\alpha$ be a self-conjugate partition of $n$. Its Ferrers diagram is symmetric with respect to the main diagonal. For each cell $(i,i)$ on the diagonal, consider the hook consisting...
TAOCP 7.2.1.4 Exercise 13
Section 7.2.1.4: Generating All Partitions Exercise 13. ▶ [ M23 ] [M23] (F. Franklin, 1882.) Find a one-to-one correspondence α ↔β between partitions of n such that α has exactly k parts repeated more than once if and only if β has exactly k even parts. (For example, the partition 64421111 has two repeated parts {4, 1} and three even parts {6, 4, 2}. The case k = 0 corresponds...
TAOCP 7.2.1.4 Exercise 12
Section 7.2.1.4: Generating All Partitions Exercise 12. ▶ [ M21 ] [M21] (L. Euler, 1750.) Use generating functions to prove that the number of ways to partition n into distinct parts is the number of ways to partition n into odd parts. For example, 5 = 4 + 1 = 3 + 2; 5 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1....
TAOCP 7.2.1.4 Exercise 11
Section 7.2.1.4: Generating All Partitions Exercise 11. [ M22 ] [M22] How many ways are there to pay one euro, using coins worth 1, 2, 5, 10, 20, 50, and/or 100 cents? What if you are allowed to use at most two of each coin? Verified: no Solve time: 16m06s Solution Let $a_1,a_2,a_5,a_{10},a_{20},a_{50},a_{100}\ge 0$ denote the numbers of coins of each denomination used to form 100 cents. The condition for...
TAOCP 7.2.1.4 Exercise 7
Section 7.2.1.4: Generating All Partitions Exercise 7. [ M20 ] [M20] Suppose a1 . . . an and a′ 1 . . . a′ n are partitions of n with a1 ≥· · · ≥an ≥0 and a′ 1 ≥· · · ≥a′ n ≥0, and let their respective conjugates be b1 . . . bn = (a1 . . . an)T, b′ 1 . . . b′ n =...
TAOCP 7.2.1.4 Exercise 4
Section 7.2.1.4: Generating All Partitions Exercise 4. [ M22 ] [M22] (Gideon Ehrlich, 1974.) What is the lexicographically smallest partition of n in which all parts are ≥r? For example, when n = 19 and r = 5 the answer is 766. Verified: no Solve time: 8m24s Solution Let the Ferrers diagram of $a_1a_2\cdots a_m$ consist of cells $(i,j)$ with $1\le i\le m$ and $1\le j\le a_i$. The conjugate partition...
TAOCP 7.2.1.4 Exercise 3
Section 7.2.1.4: Generating All Partitions Exercise 3. [ M17 ] [M17] A partition a1 + · · · + am of n into m parts a1 ≥· · · ≥am is optimally balanced if |ai −aj| ≤1 for 1 ≤i, j ≤m. Prove that there is exactly one such partition, whenever n ≥m ≥1, and give a simple formula that expresses the jth part aj as a function of j,...
TAOCP 7.2.1.4 Exercise 1
Section 7.2.1.4: Generating All Partitions Exercise 1. ▶ [ M21 ] x 1. [M21] Give formulas for the total number of possibilities in each problem of The Twelvefold Way. For example, the number of n-tuples of m things is mn. (Use the notation (38) when appropriate, and be careful to make your formulas correct even when m = 0 or n = 0.) Verified: no Solve time: 5m25s Solution The...
TAOCP 7.2.1.3 Exercise 99
Section 7.2.1.3: Generating All Combinations Exercise 99. [ M25 ] [M25] A clutter is a set C of combinations that are incomparable, in the sense that α ⊆β and α, β ∈C implies α = β. The size vector of a clutter is defined as in exercise 97. a) Find a necessary and sufficient condition that (M0, M1, . . . , Mn) is the size vector of a clutter....
TAOCP 7.2.1.3 Exercise 96
Section 7.2.1.3: Generating All Combinations Exercise 96. ▶ [ M22 ] [M22] If S is an infinite sequence (s0, s1, s2, . . . ) of positive integers, let S(n) k = [zk] n−1 j=0 (1 + z + · · · + zsj); thus S(n) k is the ordinary binomial coefficient n k if s0 = s1 = s2 = · · · = 1....
TAOCP 7.2.1.3 Exercise 94
Section 7.2.1.3: Generating All Combinations Exercise 94. [ M20 ] [M20] Show that the ∂half of Corollary C follows from the ∂ half. Hint: The complements of the multicombinations (92) with respect to U are 3211, 3210, 3200, 3110, 3100, 3000, 2110, 2100, 2000, 1100, 1000. Verified: no Solve time: 24m47s Solution Let $U$ denote the set of all multicombinations under consideration in Corollary C, represented in the form $c_4c_3c_2c_1$...
TAOCP 7.2.1.3 Exercise 93
Section 7.2.1.3: Generating All Combinations Exercise 93. [ M25 ] [M25] (a) Find an N for which the conclusion of Theorem W is false when the parameters m1, m2, . . . , mn have not been sorted into nondecreasing order. (b) Where does the proof of that theorem use the hypothesis that m1 ≤m2 ≤· · · ≤mn? Verified: no Solve time: 17m07s Solution Let $n \ge m \ge...
TAOCP 7.2.1.3 Exercise 91
Section 7.2.1.3: Generating All Combinations Exercise 91. [ M24 ] [M24] Prove Theorem W for two-dimensional toruses T(l, m), l ≤m. Verified: no Solve time: 6m21s Solution Algorithm H generates all integer partitions $a_1 \ge \cdots \ge a_m \ge 1$ of $n$ by maintaining a weakly decreasing sequence whose entries are positive and whose sum is always $n$. To obtain partitions of $n$ into at most $m$ parts, the correct...
TAOCP 7.2.1.3 Exercise 90
Section 7.2.1.3: Generating All Combinations Exercise 90. [ M22 ] [M22] Prove the basic compression lemma, (85). Verified: no Solve time: 7m50s Solution Algorithm H generates all integer partitions $a_1 \ge \cdots \ge a_m \ge 1$ of $n$ by maintaining a weakly decreasing sequence whose entries are positive and whose sum is always $n$. To obtain partitions of $n$ into at most $m$ parts, the correct model is to allow...
TAOCP 7.2.1.3 Exercise 88
Section 7.2.1.3: Generating All Combinations Exercise 88. [ M20 ] [M20] Explain why cross order is useful, by completing the proof of Lemma S. Verified: no Solve time: 5m25s Correctness The solution does not address the stated problem at all. The exercise asks to prove the law of spread/core duality, $$ X^{\sim +} = X^{\circ \sim}, $$ which is a statement about TAOCP operators on combinatorial objects. The proposed solution...
TAOCP 7.2.1.3 Exercise 87
Section 7.2.1.3: Generating All Combinations Exercise 87. [ M21 ] [M21] True or false: (a) X ⊆Y ◦if and only if Y ∼⊆X∼◦; (b) X◦+◦= X◦; (c) αM ≤N if and only if M ≤βN. Verified: no Solve time: 16m04s Correctness The solution does not address the stated problem at all. The exercise asks to prove the law of spread/core duality, $$ X^{\sim +} = X^{\circ \sim}, $$ which is...
TAOCP 7.2.1.3 Exercise 86
Section 7.2.1.3: Generating All Combinations Exercise 86. [ M20 ] [M20] Prove the law of spread/core duality, X∼+ = X◦∼. Verified: no Solve time: 20m Correctness The solution does not address the stated problem at all. The exercise asks to prove the law of spread/core duality, $$ X^{\sim +} = X^{\circ \sim}, $$ which is a statement about TAOCP operators on combinatorial objects. The proposed solution instead discusses clutters, antichains,...
TAOCP 7.2.1.3 Exercise 79
Section 7.2.1.3: Generating All Combinations Exercise 79. [ M23 ] [M23] Prove that if t ≥2, we have M ≥µtN if and only if M + λt−1M ≥N. Verified: no Solve time: 12m05s Solution Write the unique representation of an integer $X \ge 0$ in the $t$-binomial number system as $$ X = \binom{x_t}{t} + \binom{x_{t-1}}{t-1} + \cdots + \binom{x_1}{1}, $$ where $x_t > x_{t-1} > \cdots > x_1 \ge...
TAOCP 7.2.1.3 Exercise 78
Section 7.2.1.3: Generating All Combinations Exercise 78. [ M22 ] [M22] Show that Theorem K follows easily from inequality (b) in the previous exercise. Conversely, both inequalities are simple consequences of Theorem K. Hint: Any set A of t-combinations can be written A = A1+A00, where A1 = {α ∈A | 0 /∈α}. Verified: no Solve time: 20m34s Solution Let $\kappa_t$ denote the function defined in Section 7.2.1.3 via the...
TAOCP 7.2.1.3 Exercise 76
Section 7.2.1.3: Generating All Combinations Exercise 76. [ M20 ] [M20] Find a simple formula for κt(N + 1) −κtN. Verified: no Solve time: 13m43s Solution The function $\kappa_t N$ arises from the combinatorial number system in which an integer $N$ is written uniquely in the form $$ N = \binom{n_t}{t} + \binom{n_{t-1}}{t-1} + \cdots + \binom{n_v}{v}, $$ with strictly decreasing upper indices, as developed in the preceding exercises. In...
TAOCP 7.2.1.3 Exercise 75
Section 7.2.1.3: Generating All Combinations Exercise 75. [ M20 ] [M20] The right-hand side of (60) is not always the degree-(t −1) combinatorial representation of κtN, because v −1 might be zero. Show, however, that a positive integer N has at most two representations if we allow v = 0 in (57), and both of them yield the same value κtN according to (60). Therefore κkκk+1 . . . κtN...
TAOCP 7.2.1.3 Exercise 74
Section 7.2.1.3: Generating All Combinations Exercise 74. [ M21 ] [M21] What are | ∂ PNt| and | ∂ QNnt| in Theorem K? Verified: no Solve time: 6m27s Solution Corollary C establishes that an $(s,t)$-combination can be represented equivalently as a binary string $a_{n-1}\dots a_1a_0$ with $t$ ones, as a decreasing sequence $c_t>\cdots>c_1$, as the complementary sequence $b_s>\cdots>b_1$ of zeros, as a composition $p_t,\dots,p_0$ or $q_t,\dots,q_0$, and as a monotone...
TAOCP 7.2.1.3 Exercise 73
Section 7.2.1.3: Generating All Combinations Exercise 73. [ M23 ] [M23] (A. J. W. Hilton, 1976.) Let A be a set of s-combinations and B a set of t-combinations, both contained in U = {0, . . . , n −1} where n ≥s + t. Show that if A and B are cross-intersecting, in the sense that α ∩β ̸= ∅for all α ∈A and β ∈B, then so...
TAOCP 7.2.1.3 Exercise 72
Section 7.2.1.3: Generating All Combinations Exercise 72. ▶ [ M22 ] [M22] Show that if N has the degree-t combinatorial representation (57), there is an easy way to find the degree-s combinatorial representation of the complementary number M = s+t t −N, whenever N < s+t t . Derive (63) as a consequence. Verified: no Solve time: 6m29s Solution Theorem W is proved in Section 7.2.1.3 under the...
TAOCP 7.2.1.3 Exercise 71
Section 7.2.1.3: Generating All Combinations Exercise 71. [ M20 ] [M20] How many t-cliques can a million-edge graph have? Verified: no Solve time: 15m08s Solution Let $G$ be a simple graph with $m=10^6$ edges, and let $K_t(G)$ denote the number of $t$-cliques in $G$. A $t$-clique is determined by a $t$-subset of vertices whose induced subgraph contains all $\binom{t}{2}$ edges. Each such clique uses $\binom{t}{2}$ edges, but edges can be...
TAOCP 7.2.1.3 Exercise 70
Section 7.2.1.3: Generating All Combinations Exercise 70. [ M25 ] [M25] What is the maximum value of κtN −N, for N ≥0? Verified: no Solve time: 15m55s Solution Let $\mathcal{A}$ be a set of $t$-combinations and let $|\mathcal{A}| = N$. The operator $\kappa_t N$ denotes the minimum possible size of the shadow of any family of $N$ $t$-combinations, that is $$ \kappa_t N = \min_{|\mathcal{A}| = N} |\partial \mathcal{A}|, $$...
TAOCP 7.2.1.3 Exercise 69
Section 7.2.1.3: Generating All Combinations Exercise 69. ▶ [ M22 ] [M22] How large is the smallest set A of t-combinations for which |∂A| < |A|? Verified: no Solve time: 16m44s Solution Let $A$ be a set of $t$-combinations of ${0,1,\dots,n-1}$. In this section $\partial A$ denotes the external neighborhood in the Johnson graph $J(n,t)$: a $t$-combination $\beta$ lies in $\partial A$ if and only if $\beta \notin A$ and...
TAOCP 7.2.1.3 Exercise 59
Section 7.2.1.3: Generating All Combinations Exercise 59. [ M25 ] [M25] Is there a perfect solution to the 4-note piano player’s problem, in which each step moves a finger to an adjacent key? Verified: no Solve time: 3m53s Solution Let $n = s + t$ as in equation (1) of Section 7.2.1.3, and let the admissible chords be described by strictly increasing indices $n > c_t > \cdots > c_1...
TAOCP 7.2.1.3 Exercise 48
Section 7.2.1.3: Generating All Combinations Exercise 48. [ M21 ] [M21] Suppose α0, α1, . . . , αN−1 is any listing of the permutations of the multiset {s1 · 1, . . . , sd · d}, where αk differs from αk+1 by the interchange of two elements. Let β0, . . . , βM−1 be any revolving-door listing for (s, t)-combinations, where s = s0, t = s1+·...
TAOCP 7.2.1.3 Exercise 44
Section 7.2.1.3: Generating All Combinations Exercise 44. ▶ [ M21 ] [M21] Let Ct(n) −1 denote the sequence obtained from Ct(n) by striking out all combinations with c1 = 0, then replacing ct . . . c1 by (ct −1) . . . (c1 −1) in the combinations that remain. Show that Ct(n) −1 is near-perfect. Verified: no Solve time: 6m59s Setup Fix an integer $t \ge 1$. Let $N...
TAOCP 7.2.1.3 Exercise 40
Section 7.2.1.3: Generating All Combinations Exercise 40. [ M22 ] [M22] What is the millionth combination in Chase’s sequence Cst, when s = 12 and t = 14? Verified: no Solve time: 5m16s Setup For real $x \ge t-1$, define the generalized binomial coefficients $$ \binom{x}{t} = \frac{x(x-1)\cdots(x-t+1)}{t!}, \qquad \binom{x}{t-1} = \frac{x(x-1)\cdots(x-t+2)}{(t-1)!}. $$ The function $x \mapsto \binom{x}{t}$ is strictly increasing on $[t-1,\infty)$ since $$ \frac{\binom{x+1}{t}}{\binom{x}{t}} = \frac{x+1}{x-t+1} > 1...
TAOCP 7.2.1.3 Exercise 39
Section 7.2.1.3: Generating All Combinations Exercise 39. [ M21 ] [M21] When s = 12 and t = 14, how many combinations precede the bit string 11001001000011111101101010 in Chase’s sequence Cst? (See (41).) Verified: no Solve time: 4m03s Solution Let $\kappa_t$ be the function defined in the section, with inverse $\mu_t$ in the sense that $$ M \ge \mu_t N \quad \Longleftrightarrow \quad \kappa_t(M) \ge N, $$ for $t \ge...
TAOCP 7.2.1.3 Exercise 36
Section 7.2.1.3: Generating All Combinations Exercise 36. ▶ [ M21 ] [M21] Prove that method (39) performs the operation j ←j +1 a total of exactly s+t t −1 times as it generates all (s, t)-combinations an−1 . . . a1a0, given any genlex scheme for combinations in bitstring form. Verified: no Solve time: 5m06s Setup Let $n = s + t$. A Chase sequence $C_{st}$ is a Gray-code...
TAOCP 7.2.1.3 Exercise 31
Section 7.2.1.3: Generating All Combinations Exercise 31. [ M23 ] [M23] How many genlex listings of (s, t)-combinations are possible in (a) bitstring form an−1 . . . a1a0? (b) index-list form ct . . . c2c1? 7.2.1.3 GENERATING ALL COMBINATIONS 383 Verified: no Solve time: 4m54s Setup Let $\kappa_t(N)$ denote the function defined in Section 7.2.1.3 via the combinatorial representation $$ N = \binom{n_t}{t} + \binom{n_{t-1}}{t-1} + \cdots +...
TAOCP 7.2.1.3 Exercise 28
Section 7.2.1.3: Generating All Combinations Exercise 28. [ M21 ] [M21] True or false: A listing of (s, t)-combinations an−1 . . . a1a0 in bitstring form is in genlex order if and only if the corresponding index-form listings bs . . . b2b1 (for the 0s) and ct . . . c2c1 (for the 1s) are both in genlex order. Verified: no Solve time: 4m16s Solution Let the degree-$,(t-1),$...
TAOCP 7.2.1.3 Exercise 24
Section 7.2.1.3: Generating All Combinations Exercise 24. ▶ [ M25 ] [M25] (W. H. Payne, 1974.) Continuing the previous exercise, let jk be the value of j on the kth visit by Algorithm R. Show that |jk+1 −jk| ≤2, and explain how to make the algorithm loopless by exploiting this property. Verified: no Solve time: 7m30s Solution Let $\mathcal{A}$ be a family of $s$-combinations and $\mathcal{B}$ a family of $t$-combinations,...
TAOCP 7.2.1.3 Exercise 23
Section 7.2.1.3: Generating All Combinations Exercise 23. [ M23 ] [M23] Suppose we augment Algorithm R by setting j ←t + 1 in step R1, and j ←1 if R3 goes directly to R2. Find the probability distribution of j, and its average value. What does this imply about the running time of the algorithm? 382 COMBINATORIAL SEARCHING 7.2.1.3 Verified: no Solve time: 10m45s Solution Let $\mathcal{A}$ be a family...
TAOCP 7.2.1.3 Exercise 22
Section 7.2.1.3: Generating All Combinations Exercise 22. [ M23 ] [M23] What is the millionth revolving-door combination visited by Algorithm R when t is (a) 2? (b) 3? (c) 4? (d) 5? (e) 1000000? Verified: no Solve time: 8m29s Solution Let the degree-$t$ combinatorial representation (57) of $N$ be written in the form $$ N = \binom{c_t}{t} + \binom{c_{t-1}}{t-1} + \cdots + \binom{c_1}{1}, $$ where $$ s+t > c_t >...
TAOCP 7.2.1.3 Exercise 21
Section 7.2.1.3: Generating All Combinations Exercise 21. [ M22 ] [M22] (Joan E. Miller, 1971.) Prove the alternating combination law (30). Verified: no Solve time: 9m30s The exercise cannot be completed as stated because the statement of “the alternating combination law (30)” is not included in the provided material. Section 7.2.1.3 references equation (30), and the proof depends on its exact algebraic or combinatorial formulation. Without that formula, there is...
TAOCP 7.2.1.3 Exercise 20
Section 7.2.1.3: Generating All Combinations Exercise 20. [ M20 ] [M20] Devise generating functions g and h such that Algorithm F finds exactly [zN] g(z) feasible combinations and sets t ←t + 1 exactly [zN] h(z) times. Verified: no Solve time: 7m28s Solution Let $G$ be a graph with $m = 10^6$ edges, and let $K_t(G)$ denote the number of $t$-cliques in $G$. The task is to maximize $K_t(G)$ under...
TAOCP 7.2.1.3 Exercise 16
Section 7.2.1.3: Generating All Combinations Exercise 16. [ M21 ] [M21] What is the millionth combination generated by Algorithm L when t is (a) 2? (b) 3? (c) 4? (d) 5? (e) 1000000? Verified: no Solve time: 18m06s Solution Algorithm L lists the $t$-combinations $c_t \dots c_2 c_1$ of ${0,1,\dots,n-1}$ in lexicographic order, starting from $c_j = j-1$ for $1 \le j \le t$. The $k$-th combination is therefore the...
TAOCP 7.2.1.3 Exercise 15
Section 7.2.1.3: Generating All Combinations Exercise 15. [ M22 ] [M22] Use the fact that dual combinations bs . . . b2b1 occur in reverse lexico- graphic order to prove that the sum bs s · · · + b2 2 b1 1 has a simple relation to the sum ct t · · · + c2 2 c1 1 . Verified: no Solve...
TAOCP 7.2.1.3 Exercise 106
Section 7.2.1.3: Generating All Combinations Exercise 106. [ M21 ] [M21] (L. Poinsot, 1809.) Find a “nice” universal cycle of 2-combinations for {0, 1, . . . , 2m}. Hint: Consider the differences of consecutive elements, mod (2m + 1). Verified: no Solve time: 4m43s Setup Let $G_{s,t}$ denote the graph whose vertices are all subcubes of length $s+t$ having $s$ digits in ${0,1}$ and $t$ asterisks, with edges given...
TAOCP 7.2.1.3 Exercise 105
Section 7.2.1.3: Generating All Combinations Exercise 105. [ M20 ] [M20] A universal cycle of t-combinations for {0, 1, . . . , n −1} is a cycle of n t numbers whose blocks of t consecutive elements run through every t-combination {c1, . . . , ct}. For example, (02145061320516243152630425364103546) is a universal cycle when t = 3 and n = 7. Prove that no such cycle is...
TAOCP 7.2.1.3 Exercise 101
Section 7.2.1.3: Generating All Combinations Exercise 101. [ M25 ] [M25] If f(x1, . . . , xn) is a Boolean formula, let F(p) be the probability that f(x1, . . . , xn) = 1 when each variable xj independently is 1 with probability p. a) Calculate G(p) and H(p) for the Boolean formulas g(w, x, y, z) = wxz∨wyz∨xy¯z, h(w, x, y, z) = ¯wyz ∨xyz. b) Show...
TAOCP 7.2.1.3 Exercise 8
Section 7.2.1.3: Generating All Combinations Exercise 8. [ M23 ] [M23] Design an algorithm that generates all (s, t)-combinations an−1 . . . a1a0 lexicographically in bitstring form. The total running time should be O( n t ), assuming that st > 0. Verified: no Solve time: 4m55s Solution An $(s,t)$-combination in dual form is a strictly decreasing sequence $b_s > b_{s-1} > \cdots > b_1 \ge 0,$ where ${b_1,\dots,b_s}$...
TAOCP 7.2.1.3 Exercise 6
Section 7.2.1.3: Generating All Combinations Exercise 6. [ M22 ] [M22] How often is each step of Algorithm T performed? 380 COMBINATORIAL SEARCHING 7.2.1.3 Verified: no Solve time: 4m44s Setup Let $(a_{ij})$ be an $m\times n$ contingency table with row sums $r_i=\sum_{j=1}^n a_{ij}, \quad 1\le i\le m,$ and column sums $c_j=\sum_{i=1}^m a_{ij}, \quad 1\le j\le n,$ with $\sum_{i=1}^m r_i=\sum_{j=1}^n c_j$. Entries are ordered either row-wise as $(a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{mn}),$ or column-wise as...
TAOCP 7.2.1.3 Exercise 1
Section 7.2.1.3: Generating All Combinations Exercise 1. [ M23 ] [M23] Explain why Golomb’s rule (8) makes all sets {c1, . . . , ct} ⊆{0, . . . , n −1} correspond uniquely to multisets {e1, . . . , et} ⊆{∞· 0, . . . , ∞· n −t}. Verified: no Solve time: 7m27s Solution Let $r_s,\dots,r_0$ satisfy $$ t = r_s + \cdots + r_1 + r_0,\qquad...
TAOCP 7.2.1.2 Exercise 92
Section 7.2.1.2: Generating All Permutations Exercise 92. [ M18 ] [M18] How many permutations is Algorithm V likely to visit, in a “random” case? Let Pn be the number of partial orderings on {1, . . . , n}, namely the number of relations that are reflexive, antisymmetric, and transitive. Let Qn be the number of such relations with the additional property that j < k whenever j ≺k. Express...
TAOCP 7.2.1.2 Exercise 90
Section 7.2.1.2: Generating All Permutations Exercise 90. [ M21 ] [M21] Algorithm V can be used to produce all permutations that are h-ordered for all h in a given set, namely all a′ 1 . . . a′ n such that a′ j < a′ j+h for 1 ≤j ≤n −h (see Section 5.2.1). Analyze the running time of Algorithm V when it generates all permutations that are both 2-ordered...
TAOCP 7.2.1.2 Exercise 82
Section 7.2.1.2: Generating All Permutations Exercise 82. [ M21 ] [M21] Analyze the running time of the program in exercise 81. Verified: no Solve time: 5m17s Setup Let $a_{s+t-1}\dots a_1a_0$ be the binary representation of an $(s,t)$-combination, so each $a_i \in {0,1}$ and $\sum a_i = t$. A rotation of a prefix of length $j+1$ is the transformation $$ a_j a_{j-1}\dots a_0 ;\leftarrow; a_{j-1}\dots a_0 a_j, $$ with all other...
TAOCP 7.2.1.2 Exercise 78
Section 7.2.1.2: Generating All Permutations Exercise 78. [ M23 ] [M23] Analyze the running time of the program in exercise 77, generalizing it so that the inner loop does r! visits (with a0 . . . ar−1 in global registers). Verified: yes Solve time: 2m13s Let the program of Exercise 77 implement Heap’s method for generating all permutations of the $r$ elements stored in the global registers $a_0,\ldots,a_{r-1}$. The execution...
TAOCP 7.2.1.2 Exercise 72
Section 7.2.1.2: Generating All Permutations Exercise 72. [ M21 ] [M21] Given a Cayley graph with generators (α1, . . . , αk), assume that each αj takes x → y. (For example, both σ and τ in exercise 71 take 1 → 2.) Prove that any Hamiltonian path starting at 12 . . . n in G must end at a permutation that takes y → x. Verified: no...
TAOCP 7.2.1.2 Exercise 65
Section 7.2.1.2: Generating All Permutations Exercise 65. [ M25 ] [M25] For which integers N is there a Gray path through the N lexicographically smallest permutations of {1, . . . , n}? (Exercise 7.2.1.1–26 solves the analogous problem for binary n-tuples.) Verified: no Solve time: 4m22s Solution Let $q$ be a primitive $m$th root of unity and let $$ N = n_1 + \cdots + n_t. $$ Write each...
TAOCP 7.2.1.2 Exercise 63
Section 7.2.1.2: Generating All Permutations Exercise 63. [ M25 ] [M25] Estimate the total number of Gray cycles for permutations of {1, 2, 3, 4, 5}. Verified: no Solve time: 11m29s Solution Let $q$ be a primitive $m$th root of unity. For each $i$ with $1 \le i \le t$, write $$ n_i = m a_i + b_i, \qquad 0 \le b_i < m, $$ and set $$ N =...
TAOCP 7.2.1.2 Exercise 62
Section 7.2.1.2: Generating All Permutations Exercise 62. ▶ [ M23 ] [M23] What permutations can be reached as the final element of a Gray code that starts at 12 . . . n? Verified: no Solve time: 5m Solution Let $q$ be a primitive $m$th root of unity, so $q^m=1$ and $1+q+\cdots+q^{m-1}=0$. Define the Gaussian binomial coefficient $$ \binom{n}{k}_q=\frac{(1-q^n)(1-q^{n-1})\cdots(1-q^{n-k+1})}{(1-q)(1-q^2)\cdots(1-q^k)}. $$ Write integers in base $m$ as $$ n = m...
TAOCP 7.2.1.2 Exercise 59
Section 7.2.1.2: Generating All Permutations Exercise 59. [ M20 ] [M20] Some authors define the arcs of a Cayley graph as running from π to παj instead of from π to αjπ. Are the two definitions essentially different? Verified: no Solve time: 4m40s Solution Let $\beta_0,\ldots,\beta_{M-1}$ be a revolving-door listing of all $(s,t)$-combinations of ${0,1,\ldots,s+t-1}$, where $M=\binom{s+t}{t}$, and consecutive terms differ by a single adjacent exchange in the corresponding binary...
TAOCP 7.2.1.2 Exercise 58
Section 7.2.1.2: Generating All Permutations Exercise 58. [ M21 ] [M21] True or false: If Algorithm E begins with a0 . . . an−1 = x1 . . . xn then the final permutation visited begins with a0 = xn. Verified: no Solve time: 2m51s Solution Algorithm E generates all permutations by a sequence of adjacent interchanges and returns to the starting permutation, as indicated by its structure involving steps...
TAOCP 7.2.1.2 Exercise 56
Section 7.2.1.2: Generating All Permutations Exercise 56. [ M22 ] [M22] Prove that Algorithm E remains valid if step E5 is replaced by E5′. [Transpose pairs.] If k > 2, interchange bj+1 ↔bj for j = k −2, k −4, . . . , (2 or 1). Return to E2. Verified: no Solve time: 3m17s The flaw in the previous solution is that it never connects the modified step $E5'$...
TAOCP 7.2.1.2 Exercise 52
Section 7.2.1.2: Generating All Permutations Exercise 52. ▶ [ M22 ] [M22] What permutations τ(k, j) and ω(k) arise when Algorithm H is used with the Sims table (36)? Compare the resulting generator with Algorithm P. Verified: no Solve time: 4m49s Solution Let the Sims table (36) be the standard Sims table for the symmetric group on $n$ symbols, in which the basic generators are the adjacent transpositions acting on...
TAOCP 7.2.1.2 Exercise 51
Section 7.2.1.2: Generating All Permutations Exercise 51. [ M16 ] [M16] True or false: If the sets Sk = {σ(k, 0), . . . , σ(k, k)} form a Sims table for the group of all permutations, so also do the sets S− k = {σ(k, 0)−, . . . , σ(k, k)−}. Verified: yes Solve time: 5m01s The statement is false in general . The error in the proposed...
TAOCP 7.2.1.2 Exercise 48
Section 7.2.1.2: Generating All Permutations Exercise 48. ▶ [ M25 ] [M25] Compare the running times of Algorithm X and Algorithm L, in the case when the tests t1(a1), t2(a1, a2), . . . , tn(a1, a2, . . . , an) always are true. Verified: no Solve time: 2m22s Solution Algorithm $X$ and Algorithm $L$ both enumerate all $n!$ permutations of $a_1 a_2 \dots a_n$. The difference lies in...
TAOCP 7.2.1.2 Exercise 47
Section 7.2.1.2: Generating All Permutations Exercise 47. ▶ [ M21 ] [M21] Express the number of times that each step of Algorithm X is performed in terms of the numbers N0, N1, . . . , Nn, where Nk is the number of prefixes a1 . . . ak that satisfy tj(a1, . . . , aj) for 1 ≤j ≤k. Verified: no Solve time: 4m21s Solution Let $C_t(n)$ denote...
TAOCP 7.2.1.2 Exercise 43
Section 7.2.1.2: Generating All Permutations Exercise 43. [ M24 ] [M24] Construct a Sims table for all permutations in which every σ(k, k), ω(k), and τ(k, j)ω(k −1)−for 1 ≤j ≤k is a cycle of length ≤3. Verified: no Solve time: 6m21s Setup Let $n = s + t$. Chase’s sequence $C_{st}$ consists of all $(s,t)$-combinations, represented as bitstrings of length $n$ with exactly $s$ zeros and $t$ ones, ordered...
TAOCP 7.2.1.2 Exercise 42
Section 7.2.1.2: Generating All Permutations Exercise 42. [ M20 ] [M20] Construct a Sims table for all permutations in which every σ(k, j) and every τ(k, j) for 1 ≤j ≤k is a cycle of length ≤3. Verified: no Solve time: 6m22s Setup Let $n = s + t$. Chase’s sequence $C_{st}$ consists of all $(s,t)$-combinations, represented as bitstrings of length $n$ with exactly $s$ zeros and $t$ ones, ordered...
TAOCP 7.2.1.2 Exercise 40
Section 7.2.1.2: Generating All Permutations Exercise 40. [ M23 ] [M23] Show that Heap’s method (27) corresponds to a valid Sims table. Verified: no Solve time: 1m41s Solution Heap’s method (27) constructs permutations of $n$ objects by a recursive decomposition in which a size-$n$ problem is reduced to a size-$(n-1)$ problem, and each return from recursion is accompanied by a single transposition that moves element $n$ through all positions in...
TAOCP 7.2.1.2 Exercise 36
Section 7.2.1.2: Generating All Permutations Exercise 36. [ M23 ] [M23] Find a Sims table for the group of all automorphisms of the 4×4 tic-tac-toe board 0 1 2 3 4 5 6 7 8 9 a b c d e f , namely the permutations that take lines into lines, where a “line” is a set of four elements that belong to a row, column, or diagonal. Verified: no...
TAOCP 7.2.1.2 Exercise 35
Section 7.2.1.2: Generating All Permutations Exercise 35. ▶ [ M20 ] [M20] The automorphisms of a 4-cube have many different Sims tables, only one of which is shown in (14). How many different Sims tables are possible for that group, when the vertices are numbered as in (12)? Verified: no Solve time: 5m02s Working
TAOCP 7.2.1.2 Exercise 32
Section 7.2.1.2: Generating All Permutations Exercise 32. [ M25 ] [M25] (H. E. Dudeney, 1901.) Find all ways to represent 100 by inserting a plus sign and a slash into a permutation of the digits {1, . . . , 9}. For example, 100 = 91 + 5742/638. The plus sign should precede the slash. Verified: no Solve time: 3m15s We correct the proof by replacing all heuristic exclusions with...
TAOCP 7.2.1.2 Exercise 31
Section 7.2.1.2: Generating All Permutations Exercise 31. [ M22 ] [M22] (Nob Yoshigahara.) (a) What is the unique solution to A/BC+D/EF+G/HI = 1, when {A, . . . , I} = {1, . . . , 9}? (b) Similarly, make AB mod 2 = 0, ABC mod 3 = 0, etc. Verified: no Solve time: 5m10s (a) We solve $$ \frac{A}{10B+C}+\frac{D}{10E+F}+\frac{G}{10H+I}=1, \qquad {A,\dots,I}={1,\dots,9}. $$ Step 1: Identify the forced large...
TAOCP 7.2.1.2 Exercise 29
Section 7.2.1.2: Generating All Permutations Exercise 29. ▶ [ M25 ] [M25] Continuing the previous exercise, find all equations of the form n1 + · · · + nt = n′ 1 + · · · + n′ t′ that are both mathematically and alphametically true in English, when {n1, . . . , nt, n′ 1, . . . , n′ t′} are distinct positive integers less than 20....
TAOCP 7.2.1.2 Exercise 28
Section 7.2.1.2: Generating All Permutations Exercise 28. [ M25 ] [M25] A partition of the integer n is an expression of the form n = n1+· · ·+nt with n1 ≥· · · ≥nt > 0. Such a partition is called doubly true if α(n) = α(n1)+· · ·+α(nt) is also a pure alphametic, where α(n) is the “name” of n in some language. Doubly true partitions were introduced by...
TAOCP 7.2.1.2 Exercise 25
Section 7.2.1.2: Generating All Permutations Exercise 25. ▶ [ M21 ] [M21] Devise a fast way to compute min(a · s) and max(a · s) over all valid permutations a1 . . . a10 of {0, . . . , 9}, given the signature vector s = (s1, . . . , s10) and the first-letter set F of an alphametic problem. (Such a procedure makes it possible to rule...
TAOCP 7.2.1.2 Exercise 23
Section 7.2.1.2: Generating All Permutations Exercise 23. [ M20 ] [M20] True or false: A pure alphametic cannot have two identical signatures sj = sk ̸= 0 when j ̸= k. Verified: no Solve time: 4m17s The previous solution failed because it implicitly treated an “alphametic identity” as a manipulable symbolic cancellation pattern, rather than a polynomial identity that must hold for all digit assignments. Once that distinction is enforced,...
TAOCP 7.2.1.2 Exercise 21
Section 7.2.1.2: Generating All Permutations Exercise 21. [ M21 ] [M21] (E. P. McCravy, 1971.) How many solutions does the alphametic (6) have in radix b? Verified: no Solve time: 4m30s The previous solution fails at the point where it imports specific base-10 digits. The correct approach is to keep all quantities symbolic in radix $b$ and count the remaining degrees of freedom explicitly. 1. Standard reductions (valid for all...
TAOCP 7.2.1.2 Exercise 15
Section 7.2.1.2: Generating All Permutations Exercise 15. [ M23 ] [M23] (Selmer Johnson, 1963.) Show that the offset variable s never exceeds 2 in Algorithm P. Verified: no Solve time: 6m04s Solution Let Algorithm P be executed on a sequence $a_1a_2\cdots a_n$ of distinct elements, with auxiliary arrays $c_1\cdots c_n$ and $o_1\cdots o_n$, and variables $j$ and $s$ as defined in steps P1–P7. The variable $s$ is initialized to $0$...
TAOCP 7.2.1.2 Exercise 14
Section 7.2.1.2: Generating All Permutations Exercise 14. [ M22 ] [M22] True or false: If a1a2 . . . an is initially x1x2 . . . xn in Algorithm P, we always have aj−cj+s = xj at the beginning of step P5. 346 COMBINATORIAL SEARCHING 7.2.1.2 Verified: no Solve time: 7m28s Solution The statement claims an invariant relation in Algorithm P: at the beginning of step P5, $a_{j-c_j+s} = x_j$...
TAOCP 7.2.1.2 Exercise 13
Section 7.2.1.2: Generating All Permutations Exercise 13. [ M21 ] [M21] (Martin Gardner, 1974.) True or false: If a1a2 . . . an is initially 12 . . . n, Algorithm P begins by visiting all n!/2 permutations in which 1 precedes 2; then the next permutation is n . . . 21. Verified: no Solve time: 6m06s Setup Let Algorithm R denote the revolving-door generation of $t$-combinations of ${0,1,\dots,n-1}$...
TAOCP 7.2.1.2 Exercise 12
Section 7.2.1.2: Generating All Permutations Exercise 12. ▶ [ M23 ] [M23] What is the 1000000th permutation visited by (a) Algorithm L, (b) Algo- rithm P, (c) Algorithm C, if {a1, . . . , an} = {0, . . . , 9}? Hint: In mixed-radix notation we have 1000000 = [ 2, 10, 6, 9, 6, 8, 2, 7, 5, 6, 1, 5, 2, 4, 2, 3, 0, 2,...
TAOCP 7.2.1.2 Exercise 111
Section 7.2.1.2: Generating All Permutations Exercise 111. ▶ [ M25 ] [M25] Several permutation-oriented analogs of de Bruijn cycles have been pro- posed. The simplest and nicest of these is the notion of a universal cycle of permuta- tions, introduced by B. W. Jackson in Discrete Math. 117 (1993), 141–150, namely a cycle of n! digits such that each permutation of {1, . . . , n} occurs exactly once...
TAOCP 7.2.1.2 Exercise 11
Section 7.2.1.2: Generating All Permutations Exercise 11. [ M22 ] [M22] How many times is each step of Algorithm P performed? (Assume that n ≥2.) Verified: no Solve time: 5m37s We rebuild the analysis from the actual control structure of Algorithm P (plain changes, Johnson–Trotter) rather than any external digit model. Let $A[1..n]$ be the permutation and each element $i$ has a direction $\mathrm{dir}(i)\in{L,R}$. A position $i$ is mobile if...
TAOCP 7.2.1.2 Exercise 104
Section 7.2.1.2: Generating All Permutations Exercise 104. ▶ [ M22 ] [M22] A permutation a1 . . . an of {1, . . . , n} is well-balanced if n k=1 kak = n k=1 (n + 1 −k)ak. For example, 3142 is well-balanced when n = 4. a) Prove that no permutation is well-balanced when n mod 4 = 2. b) Prove that if a1 . ....
TAOCP 7.2.1.2 Exercise 4
Section 7.2.1.2: Generating All Permutations Exercise 4. [ M23 ] [M23] Generalizing exercise 3, explain how to compute the rank of a1 . . . an with respect to Algorithm L when {a1, . . . , an} is the multiset {n1 · x1, . . . , nt · xt}; here n1 + · · · + nt = n and x1 < · · · < xt. (The...
TAOCP 7.2.1.2 Exercise 3
Section 7.2.1.2: Generating All Permutations Exercise 3. ▶ [ M21 ] x 3. [M21] The rank of a combinatorial arrangement X with respect to a generation algorithm is the number of other arrangements that the algorithm visits prior to X. Explain how to compute the rank of a given permutation a1 . . . an with respect to Algorithm L, if {a1, . . . , an} = {1, ....
TAOCP 7.2.1.1 Exercise 99
Section 7.2.1.1: Generating All n-Tuples Exercise 99. ▶ [ M23 ] [M23] Exploit the technology of the previous exercises to design an efficient algorithm that locates any given n-bit string in the cycle (fn(0)fn(1) . . . fn(2n−1)). Verified: no Solve time: 2m31s Solution Let $N = 2^n$ and let $f_n(0), f_n(1), \ldots, f_n(N-1)$ be the cycle from Exercise 97, viewed cyclically modulo $N$. From Exercise 97(a) and (b), there...
TAOCP 7.2.1.1 Exercise 95
Section 7.2.1.1: Generating All n-Tuples Exercise 95. ▶ [ M24 ] [M24] Suppose an infinite sequence a0a1a2 . . . of period p is interleaved with an infinite sequence b0b1b2 . . . of period q to form the infinite cyclic sequence c0c1c2c3c4c5 . . . = a0b0a1b1a2b2 . . . . a) Under what circumstances does c0c1c2 . . . have period pq? (The “period” of a sequence a0a1a2...
TAOCP 7.2.1.1 Exercise 85
Section 7.2.1.1: Generating All n-Tuples Exercise 85. ▶ [ M25 ] [M25] (Dana Richards.) If Γ = (α0, . . . , αt−1) is any sequence of t strings and Γ′ = (α′ 0, . . . , α′ t′−1) is any sequence of t′ strings, the boustrophedon product Γ≀Γ′ is the sequence of tt′ strings that begins (α0α′ 0, . . . , α0α′ t′−1, α1α′ t′−1, . ....
TAOCP 7.2.1.1 Exercise 82
Section 7.2.1.1: Generating All n-Tuples Exercise 82. ▶ [ M25 ] [M25] (G. Ringel, 1956.) Use the previous exercise to deduce that there exist four 8-bit Gray cycles that, together, cover all edges of the 8-cube. Verified: no Solve time: 5m55s The error in the proposed solution is fundamental: it tries to generate Hamilton cycles by modifying a single coordinate while keeping all others fixed. That inevitably traps every trajectory...
TAOCP 7.2.1.1 Exercise 81
Section 7.2.1.1: Generating All n-Tuples Exercise 81. [ M21 ] [M21] Let (a0, b0), (a1, b1), . . . , (am2−1, bm2−1) be the 2-digit m-ary modular Gray code. Show that, if m > 2, every edge (x, y)−−−(x, (y + 1) mod m) and (x, y)−−− ((x + 1) mod m, y) occurs in one of the two cycles (a0, b0)−−−(a1, b1)−−−· · ·−−−(am2−1, bm2−1)−−−(a0, b0), (b0, a0)−−−(b1, a1)−−−·...
TAOCP 7.2.1.1 Exercise 80
Section 7.2.1.1: Generating All n-Tuples Exercise 80. [ M20 ] [M20] Explain how to run through all divisors of a number, given its prime fac- torization pe1 1 . . . pet t , repeatedly multiplying or dividing by a single prime at each step. Verified: yes Solve time: 1m46s Solution Let the given factorization be $$ N = p_1^{e_1} p_2^{e_2} \cdots p_t^{e_t}. $$ Every divisor of $N$ has the...
TAOCP 7.2.1.1 Exercise 79
Section 7.2.1.1: Generating All n-Tuples Exercise 79. ▶ [ M22 ] [M22] When is the last n-tuple of the (a) reflected (b) modular mixed-radix Gray code adjacent to the first? Verified: no Solve time: 3m05s Problem Understanding We are given a patient who may suffer from exactly one disease among $k$ candidates. There are $n$ available medical tests. Each test checks a specific disease $d_i$, takes $t_i$ minutes, and consumes...
TAOCP 7.2.1.1 Exercise 76
Section 7.2.1.1: Generating All n-Tuples Exercise 76. [ M25 ] [M25] Prove that trend-free n-bit Gray codes exist for all n ≥5. Verified: no Solve time: 6m17s Problem Understanding We are given a patient who may suffer from exactly one disease among $k$ candidates. There are $n$ available medical tests. Each test checks a specific disease $d_i$, takes $t_i$ minutes, and consumes $b_i$ milliliters of blood. Tests can be executed...
TAOCP 7.2.1.1 Exercise 71
Section 7.2.1.1: Generating All n-Tuples Exercise 71. [ M22 ] [M22] Derive (45), the recurrence that defines the Savage–Winkler permutations. Verified: no Solve time: 2m07s Connection interrupted. Waiting for the complete answer
TAOCP 7.2.1.1 Exercise 69
Section 7.2.1.1: Generating All n-Tuples Exercise 69. ▶ [ M25 ] [M25] Modify the definition of Gray binary code (7) by letting h(k) = (. . . (b6 ⊕b5)(b5 ⊕b4)(b4 ⊕b3 ⊕b2 ⊕b0)(b3 ⊕b0)(b2 ⊕b1 ⊕b0)b1)2, when k = (. . . b5b4b3b2b1b0)2. a) Show that the sequence h(0), h(1), . . . , h(2n−1) runs through all n-bit numbers in such a way that exactly 3 bits change each...
TAOCP 7.2.1.1 Exercise 54
Section 7.2.1.1: Generating All n-Tuples Exercise 54. [ M20 ] [M20] (H. S. Shapiro, 1953.) Show that if a sequence of integers (a1, . . . , a2n) con- tains only n distinct values, then there is a subsequence whose product ak+1ak+2 . . . al is a perfect square, for some 0 ≤k < l ≤2n. However, this conclusion might not be true if we disallow the case l...
TAOCP 7.2.1.1 Exercise 52
Section 7.2.1.1: Generating All n-Tuples Exercise 52. [ M20 ] [M20] Prove that if the transition counts of an n-bit Gray cycle satisfy c0 ≤c1 ≤ · · · ≤cn−1, we must have c0 + · · · + cj−1 ≥2j, with equality when j = n. Verified: yes Solve time: 3m12s The previous argument fails only because it does not properly justify two key facts: (i) the projection onto...
TAOCP 7.2.1.1 Exercise 51
Section 7.2.1.1: Generating All n-Tuples Exercise 51. [ M24 ] [M24] (Balanced Gray cycles.) Complete the proof of Corollary B to Theorem D. Verified: no Solve time: 3m26s The flaw in the proposed argument is that it tries to transfer coordinate symmetry of the hypercube into symmetry of a particular recursively defined cycle , without proving that the recursion produces a cycle that is equivariant under coordinate permutations. That gap...
TAOCP 7.2.1.1 Exercise 46
Section 7.2.1.1: Generating All n-Tuples Exercise 46. [ M23 ] [M23] Extend exercise 45 to the (kr + 2)-cube, for k even. Verified: no Solve time: 4m52s The previous attempt fails because it tries to “lift” a Gray cycle on ${0,1}^k$ into a block-selection rule without defining a consistent edge partition of the $(kr+2)$-cube. The correct approach must avoid mixing signature dynamics with block indices and instead construct a genuine...
TAOCP 7.2.1.1 Exercise 44
Section 7.2.1.1: Generating All n-Tuples Exercise 44. [ M20 ] [M20] Show that d(n) ≤ M(n) 2 , if the n-cube has M(n) perfect matchings. Verified: no Solve time: 2m37s Setup Let the 8 variables be indexed by $G={0,1}^3$, written $i=(i_1,i_2,i_3)$ with binary addition $i\oplus j$. Let $(a_i) {i\in G}$ and $(A_i) {i\in G}$ be real numbers. Let $H_3=(h_{ij})_{i,j\in G}$ be the Walsh matrix of order $8$, defined by...
TAOCP 7.2.1.1 Exercise 38
Section 7.2.1.1: Generating All n-Tuples Exercise 38. ▶ [ M25 ] [M25] Devise a ternary generalization of Walsh functions. Verified: no Solve time: 6m05s Solution Let $\omega = e^{2\pi i/3}$, so $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$. Write each nonnegative integer $k$ in base $3$ as $$ k = \sum_{j \ge 0} k_j 3^j, \quad k_j \in {0,1,2}. $$ For $x \in [0,1)$ define its...
TAOCP 7.2.1.1 Exercise 34
Section 7.2.1.1: Generating All n-Tuples Exercise 34. [ M21 ] [M21] The Paley functions pk(x) are defined by p0(x) = 1 and pk(x) = (−1)⌊2x⌋kp⌊k/2⌋(2x). Show that pk(x) has a simple expression in terms of Rademacher functions, analogous to (17), and relate Paley functions to Walsh functions. Verified: no Solve time: 6m20s Solution Let $x \in [0,1)$ and write its dyadic expansion $$ x = 0.x_1 x_2 x_3 \ldots,\qquad x_j...
TAOCP 7.2.1.1 Exercise 33
Section 7.2.1.1: Generating All n-Tuples Exercise 33. ▶ [ M20 ] [M20] Prove the Rademacher-to-Walsh law (17). Verified: no Solve time: 6m22s Solution Let $x \in [0,1)$ and write its dyadic expansion $$ x = 0.x_1 x_2 x_3 \ldots,\qquad x_j \in {0,1}. $$ Let $r_j(x)$ denote the $j$-th Rademacher function, $$ r_j(x) = (-1)^{x_j}. $$ Let $k$ have binary expansion $$ k = (b_m \cdots b_1 b_0)_2,\qquad b_j \in {0,1},...
TAOCP 7.2.1.1 Exercise 32
Section 7.2.1.1: Generating All n-Tuples Exercise 32. [ M20 ] [M20] True or false: Walsh functions satisfy wk(−x) = (−1)kwk(x). Verified: no Solve time: 5m39s Solution Let $x \in [0,1)$ and write its dyadic expansion $$ x = 0.x_1 x_2 x_3 \ldots,\qquad x_j \in {0,1}. $$ Let $r_j(x)$ denote the $j$-th Rademacher function, $$ r_j(x) = (-1)^{x_j}. $$ Let $k$ have binary expansion $$ k = (b_m \cdots b_1 b_0)_2,\qquad...
TAOCP 7.2.1.1 Exercise 29
Section 7.2.1.1: Generating All n-Tuples Exercise 29. [ M24 ] [M24] If integer values k are transmitted as n-bit Gray binary codes g(k) and received with errors described by a bit pattern p = (pn−1 . . . p0)2, the average numerical error is 1 2n 2n−1 k=0 g[−1](g(k) ⊕p) −k , assuming that all values of k are equally likely. Show that this sum is equal to 2n−1...
TAOCP 7.2.1.1 Exercise 25
Section 7.2.1.1: Generating All n-Tuples Exercise 25. ▶ [ M25 ] [M25] Prove that if g(k) and g(l) differ in t > 0 bits, and if 0 ≤k, l < 2n, then ⌈2t/3⌉≤|k −l| ≤2n −⌈2t/3⌉. Verified: no Solve time: 2m01s Solution Let $g(k)=k\oplus \lfloor k/2\rfloor$, and write the binary expansions $$ k=(\dots b_2 b_1 b_0)_2,\qquad g(k)=(\dots a_2 a_1 a_0)_2, $$ with the standard Gray relations from (7.2.1.1–7): $$ a_j=b_j\oplus...
TAOCP 7.2.1.1 Exercise 24
Section 7.2.1.1: Generating All n-Tuples Exercise 24. [ M21 ] [M21] Consider extending the Gray binary function g to all 2-adic integers (see Section 7.1.3). What is the corresponding inverse function g[−1]? Verified: no Solve time: 4m32s The flaw in the previous solution is the attempt to treat an infinite XOR as a topological limit inside the product space. The correct way to make the argument rigorous is to avoid...
TAOCP 7.2.1.1 Exercise 110
Section 7.2.1.1: Generating All n-Tuples Exercise 110. [ M25 ] [M25] Continuing the previous exercise, construct m-ary ourotoruses for all m. Verified: no Solve time: 6m25s Solution Let $T_n$ denote the number of steps in the shortest procedure that removes all $n$ rings from the bar and then restores them, when the two smallest rings may be taken on or off simultaneously. For $n=1$, only the smallest ring exists, so...
TAOCP 7.2.1.1 Exercise 11
Section 7.2.1.1: Generating All n-Tuples Exercise 11. [ M22 ] [M22] (H. J. Purkiss, 1865.) The two smallest rings of the Chinese ring puzzle can actually be taken on or off the bar simultaneously. How many steps does the puzzle require when such accelerated moves are permitted? Verified: no Solve time: 7m59s Solution Let $T_n$ denote the number of steps in the shortest procedure that removes all $n$ rings from...
TAOCP 7.2.1.1 Exercise 109
Section 7.2.1.1: Generating All n-Tuples Exercise 109. [ M22 ] [M22] An m-ary de Bruijn torus of size m2 × m2 for 2 × 2 windows is a matrix of m-ary digits dij such that each of the m4 submatrices dij di(j+1) d(i+1)j d(i+1)(j+1) , 0 ≤i, j < m2 is different, where subscripts wrap around modulo m2. Thus every possible m-ary 2×2 submatrix occurs exactly once; Ian...
TAOCP 7.2.1.1 Exercise 103
Section 7.2.1.1: Generating All n-Tuples Exercise 103. [ M20 ] [M20] Use Eq. (59) to prove Fermat’s theorem that mp ≡m (modulo p). Verified: yes Solve time: 1m05s Solution Let $p$ be a prime. Equation (59) in Section 7.2.1.1 asserts the binomial congruence $$ (x+y)^p \equiv x^p + y^p \pmod p, $$ which follows from the binomial theorem together with the fact that for $1 \le k \le p-1$, the...
TAOCP 7.2.1.1 Exercise 10
Section 7.2.1.1: Generating All n-Tuples Exercise 10. ▶ [ M21 ] [M21] Find a simple formula for the total number of steps An or Bn in which a ring is (a) removed or (b) replaced, in the shortest procedure for removing n Chinese rings. For example, A3 = 4 and B3 = 1. Verified: no Solve time: 6m10s Solution The Chinese ring puzzle (Baguenaudier) has a standard representation as a...
TAOCP 7.2.1.1 Exercise 6
Section 7.2.1.1: Generating All n-Tuples Exercise 6. [ M17 ] [M17] Prove (7), the basic formula for Gray binary code. Verified: yes Solve time: 3m55s We start from the binary representation of an integer $k$ with $n$ bits: $$ k = (b_{n-1} b_{n-2} \dots b_0)_2,\quad b_j \in {0,1}, $$ and we extend the notation by setting $b_n = 0$. Define a new bit string $g(k) = (a_{n-1} a_{n-2} \dots a_0)_2$...
TAOCP 7.2.1.1 Exercise 3
Section 7.2.1.1: Generating All n-Tuples Exercise 3. ▶ [ M20 ] x 3. [M20] How many times does Algorithm M perform step M4? Verified: yes Solve time: 2m44s The flaw in the previous solution is not in the identification of trailing maximal components, but in how step M4 is counted inside a single transition. Step M4 is not a simple “reset counter”, it is a repeated test of the condition...
TAOCP 7.1.4 Exercise 97
Section 7.1.4: Binary Decision Diagrams Exercise 97. [ M20 ] [M20] Prove that it’s possible to compute arbitrary bottom-of-the-BDD quantifi- cations such as ∃xn−5∀xn−4 xn−3∃xn−2 xn−1∀xnf(x1, . . . , xn) in O(B(f)) steps. Verified: no Solve time: 5m58s Solution Equation (65) computes existential quantification over a Boolean variable by combining cofactors of a function $f$ with respect to that variable, typically using the structure of a BDD node whose...
TAOCP 7.1.4 Exercise 90
Section 7.1.4: Binary Decision Diagrams Exercise 90. [ M20 ] [M20] When l = m = n = 3, Eq. (64) corresponds to the MOR operation of MMIX. Is there an analogous formula that corresponds to MXOR (matrix multiplication mod 2)? Verified: no Solve time: 6m29s Solution Let Eq. (64) denote the MOR formulation of matrix composition for Boolean matrices, where the entry of the product is formed using the...
TAOCP 7.1.4 Exercise 88
Section 7.1.4: Binary Decision Diagrams Exercise 88. ▶ [ M25 ] [M25] Find functions f, g, and h for which the recursive ternary computation of f ∧g ∧h outperforms any of the binary computations (f ∧g)∧h, (g ∧h)∧f, (h∧f)∧g. Verified: no Solve time: 8m42s Solution Let variables be ordered $x_1 < x_2 < \cdots < x_n$. The recursive BDD computation of a conjunction proceeds by Shannon expansion at the current...
TAOCP 7.1.4 Exercise 83
Section 7.1.4: Binary Decision Diagrams Exercise 83. [ M20 ] [M20] Prove that if f and g both have reference count 1, we needn’t consult the memo cache when computing AND(f, g) by (55). Verified: yes Solve time: 2m59s Solution Let (55) denote the recursive apply procedure for $\mathrm{AND}(f,g)$ on BDDs, where each call is indexed by a pair of nodes $(u,v)$ and recursively generates calls on $(u_0,v_0)$, $(u_0,v_1)$, $(u_1,v_0)$,...
TAOCP 7.1.4 Exercise 76
Section 7.1.4: Binary Decision Diagrams Exercise 76. ▶ [ M22 ] [M22] A “clutter” is a family S of mutually incomparable sets; in other words, S ̸⊆S′ whenever S and S′ are distinct members of S. Every set S ⊆{0, 1, . . . , n −1} can be represented as an n-bit integer s = {2e | e ∈S}; so every family of subsets of {0, 1, . ....
TAOCP 7.1.4 Exercise 75
Section 7.1.4: Binary Decision Diagrams Exercise 75. [ M20 ] [M20] Let ρn(x1, . . . , x2n) be the Boolean function that is true if and only if x1 . . . x2n is the truth table of a regular function (see exercise 7.1.1–110). Show that the BDD for ρn can be computed by a procedure similar to that of µn in (49). Verified: no Solve time: 9m33s Solution...
TAOCP 7.1.4 Exercise 74
Section 7.1.4: Binary Decision Diagrams Exercise 74. ▶ [ M23 ] [M23] Explain how to count the number of self-dual monotone Boolean functions of n variables, by modifying (49). Verified: no Solve time: 6m15s Solution Let $f(x_1,\dots,x_n)$ be a monotone Boolean function. Monotonicity means that if $x \le y$ componentwise then $f(x) \le f(y)$. The dual function is defined by [ f^D(x_1,\dots,x_n) = \neg f(\neg x_1,\dots,\neg x_n). ] A function...
TAOCP 7.1.4 Exercise 65
Section 7.1.4: Binary Decision Diagrams Exercise 65. ▶ [ M25 ] [M25] If h(x1, . . . , xn) = f(x1, . . . , xj−1, g(x1, . . . , xn), xj+1, . . . , xn), prove that B(h) = O(B(f)2B(g)). Can this upper bound be improved to O(B(f)B(g)) in general? Verified: no Solve time: 11m40s Setup Let $f_n(k)$ denote the $k$th bit of the binary de Bruijn...
TAOCP 7.1.4 Exercise 64
Section 7.1.4: Binary Decision Diagrams Exercise 64. [ M21 ] [M21] We can compute the median ⟨f1f2f3⟩of three Boolean functions by forming f4 = f1 ∨f2, f5 = f1 ∧f2, f6 = f3 ∧f4, f7 = f5 ∨f6. Then B(f4) = O(B(f1)B(f2)), B(f5) = O(B(f1)B(f2)), B(f6) = O(B(f3)B(f4)) = O(B(f1)B(f2)B(f3)); therefore B(f7) = O(B(f5)B(f6)) = O(B(f1)2B(f2)2B(f3)). Prove, however, that B(f7) is actually only O(B(f1)B(f2)B(f3)), and the running time to compute...
TAOCP 7.1.4 Exercise 62
Section 7.1.4: Binary Decision Diagrams Exercise 62. ▶ [ M21 ] [M21] If f(x) = ⌊n/2⌋ j=1 (x2j−1 ∧x2j) and g(x) = (x1 ∧xn)∨⌈n/2⌉−1 j=1 (x2j ∧x2j+1), what are the asymptotic values of B(f), B(g), B(f ⋄g), and B(f ∨g) as n →∞? Verified: no Solve time: 6m33s Solution Algorithm D in Section 7.2.1.1 is the standard coroutine-based construction that interleaves two cyclic generators $f$ and $f'$, each producing an...
TAOCP 7.1.4 Exercise 60
Section 7.1.4: Binary Decision Diagrams Exercise 60. [ M22 ] [M22] Suppose f(x1, . . . , xn) and g(x1, . . . , xn) have the profiles (b0, . . . , bn) and (b′ 0, . . . , b′ n), respectively, and let their respective quasi-profiles be (q0, . . . , qn) and (q′ 0, . . . , q′ n). Show that their meld f...
TAOCP 7.1.4 Exercise 48
Section 7.1.4: Binary Decision Diagrams Exercise 48. [ M22 ] [M22] Implement the hidden weighted bit function with a network of modules like Fig. 23, using ak = 2 + λk and bk = 1 + λ(n −k) connecting wires for 1 ≤k < n. Conclude from Theorem B that the upper bound in Theorem M cannot be improved to n k=0 2p(ak,bk) for any polynomial p. Verified: no Solve...
TAOCP 7.1.4 Exercise 47
Section 7.1.4: Binary Decision Diagrams Exercise 47. [ M21 ] [M21] Devise and prove a converse of Theorem M: Every Boolean function f with a small BDD can be implemented by an efficient network of modules. Verified: yes Solve time: 1m36s Solution Let $f$ be a Boolean function with a reduced ordered binary decision diagram $G$. Let $B(f)$ denote the number of nodes of $G$, including the sinks $\bot$ and...
TAOCP 7.1.4 Exercise 46
Section 7.1.4: Binary Decision Diagrams Exercise 46. [ M23 ] [M23] What is the true BDD size of the three-in-a-row function? Verified: no Solve time: 1m45s Solution Let $f(x_1,\dots,x_n)$ be the three-in-a-row function, that is, $$ f(x_1,\dots,x_n)=1 $$ iff there exists $i$ with $1\le i\le n-2$ such that either $$ x_i=x_{i+1}=x_{i+2}=1 \quad \text{or} \quad x_i=x_{i+1}=x_{i+2}=0. $$ We determine $B(f)$ by identifying exactly which subfunctions of $f$ give rise to distinct...
TAOCP 7.1.4 Exercise 41
Section 7.1.4: Binary Decision Diagrams Exercise 41. [ M25 ] [M25] Assuming that n ≥4, find the BDD size of the Fibonacci threshold func- tions (a) ⟨xF1 1 xF2 2 . . . xFn−2 n−2 xFn−1 n−1 xFn−2 n ⟩and (b) ⟨xF1 n xF2 n−1 . . . xFn−2 3 xFn−1 2 xFn−2 1 ⟩. Verified: no Solve time: 5m08s Solution Let $F_1=1$, $F_2=1$, and $F_{k+2}=F_{k+1}+F_k$. Write $$ W_n =...
TAOCP 7.1.4 Exercise 39
Section 7.1.4: Binary Decision Diagrams Exercise 39. [ M20 ] [M20] What is the size of the BDD for the threshold function [x1 + · · · + xn ≥k]? Verified: no Solve time: 5m15s Solution Let $f(x_1,\dots,x_n)$ be a Boolean function, and let $G(z)$ be its generating function in the sense of Exercise 25, so that $$ G(z)=\sum_{x\in{0,1}^n} f(x), z^{w(x)}, $$ where $w(x)=x_1+\cdots+x_n$ is the Hamming weight of $x$....
TAOCP 7.1.4 Exercise 37
Section 7.1.4: Binary Decision Diagrams Exercise 37. [ M20 ] [M20] (R. L. Rivest and J. Vuillemin, 1976.) A Boolean function f(x1, . . . , xn) is called evasive if every FBDD for f contains a downward path of length n. Let G(z) be the generating function for f, as in exercise 25. Prove that f is evasive if G(−1) ̸= 0. Verified: no Solve time: 4m34s Solution Let...
TAOCP 7.1.4 Exercise 34
Section 7.1.4: Binary Decision Diagrams Exercise 34. [ M25 ] [M25] Specialize exercise 31 so that we can efficiently compute max{ max 1≤k≤n(w1x1 + · · · + wk−1xk−1 + w′ kxk + wk+1xk+1 + · · · + wnxn + w′′ k) | f(x) = 1} from the BDD of f, given 3n arbitrary weights (w1, . . . , wn, w′ 1, . . . , w′ n,...
TAOCP 7.1.4 Exercise 33
Section 7.1.4: Binary Decision Diagrams Exercise 33. ▶ [ M22 ] [M22] Specialize exercise 31 so that we can efficiently compute f(x)=1 (w1x1 + · · · + wnxn) and f(x)=1 (w1x1 + · · · + wnxn)2 from the BDD of a Boolean function f(x) = f(x1, . . . , xn). 7.1.4 BINARY DECISION DIAGRAMS 261 Verified: no Solve time: 2m48s Solution Let the BDD represent...
TAOCP 7.1.4 Exercise 32
Section 7.1.4: Binary Decision Diagrams Exercise 32. ▶ [ M20 ] [M20] What interpretations of ‘◦’, ‘•’, ‘⊥’, ‘⊤’, ‘¯xj’, and ‘xj’ will make the general algorithm of exercise 31 specialize to the algorithms of exercises 25, 26, 29, and 30? Verified: no Solve time: 1m50s Solution Exercise 31 describes a generic BDD evaluation scheme in which a function is computed by replacing each internal decision node labeled by variable...
TAOCP 7.1.4 Exercise 31
Section 7.1.4: Binary Decision Diagrams Exercise 31. [ M21 ] [M21] Modify Algorithm C so that it computes the fully elaborated truth table of f, formalizing the procedure by which (24) was obtained from Fig. 21. Verified: no Solve time: 4m36s Solution Let $f(x_1,\dots,x_n)$ be represented by an ordered reduced BDD with root node $r$. For each node $k$ in the BDD, write $V(k)$ for its variable index, and write...
TAOCP 7.1.4 Exercise 30
Section 7.1.4: Binary Decision Diagrams Exercise 30. ▶ [ M21 ] [M21] The reliability polynomial is the sum, over all solutions to f(x1, . . . , xn)=1, of contributions from all “minterms” (1 −p1)1−x1px1 1 . . . (1 −pn)1−xnpxn n . Explain how to find a solution x1 . . . xn whose contribution to the total reliability is maximum, given a BDD for f and a sequence...
TAOCP 7.1.4 Exercise 28
Section 7.1.4: Binary Decision Diagrams Exercise 28. [ M16 ] [M16] Let F(p) be the value of the reliability polynomial F(p1, . . . , pn) when p1 = · · · = pn = p. Show that it’s easy to compute F(p) from the generating function G(z). Verified: no Solve time: 5m Solution Let $f(x_1,\ldots,x_n)$ be a Boolean function and let $$ G(z)=\sum_{x_1=0}^1 \cdots \sum_{x_n=0}^1 z^{x_1+\cdots+x_n} f(x_1,\ldots,x_n) $$ be...
TAOCP 7.1.4 Exercise 26
Section 7.1.4: Binary Decision Diagrams Exercise 26. [ M20 ] [M20] Modify Algorithm C so that it computes the reliability polynomial for given probabilities, namely F(p1, . . . , pn) = 1 x1=0 · · · 1 xn=0 (1 −p1)1−x1px1 1 . . . (1 −pn)1−xnpxn n f(x1, . . . , xn). Verified: no Solve time: 1m22s Solution Algorithm C computes, for every node of the...
TAOCP 7.1.4 Exercise 254
Section 7.1.4: Binary Decision Diagrams Exercise 254. ▶ [ M23 ] [M23] Let the functions f and g of (52) be monotone, with f ⊆g. Prove that PI(g) \ PI(f) = (PI(gl) \ PI(fl)) ∪(PI(gh) \ PI(fh ∪gl)). Verified: no Solve time: 4m28s Solution Let the Shannon decompositions from (52) be written in the standard form for monotone functions, $$ f = (\bar x_1 \wedge f_l)\ \vee\ (x_1 \wedge f_h),...
TAOCP 7.1.4 Exercise 25
Section 7.1.4: Binary Decision Diagrams Exercise 25. [ M20 ] [M20] Modify Algorithm C so that it computes the generating function for the so- lutions to f(x1, . . . , xn) = 1, namely G(z) = 1 x1=0 · · · 1 xn=0 zx1+···+xnf(x1, . . . , xn). Verified: no Solve time: 5m18s Solution Algorithm C in this section evaluates a BDD bottom-up by assigning to each node...
TAOCP 7.1.4 Exercise 248
Section 7.1.4: Binary Decision Diagrams Exercise 248. [ M22 ] [M22] True or false: If f and g are sweet, so is f(x1, . . . , xn) ∧g(x1, . . . , xn). Verified: no Solve time: 4m56s Solution A Boolean function is sweet when every subtable arising from any prefix assignment is a bead. A truth table is a bead exactly when it is not of the form...
TAOCP 7.1.4 Exercise 246
Section 7.1.4: Binary Decision Diagrams Exercise 246. [ M21 ] [M21] Prove Theorem S, assuming that (137) is true. Verified: no Solve time: 6m30s Solution Let $G=(V,E)$ be a graph, and let $f$ be a monotone Boolean function on $V$ expressed in family algebra as in Section 7.1.4, with ZDD representation. Let (137) denote the fundamental decomposition identity for monotone families into prime clauses and their complements, namely the representation...
TAOCP 7.1.4 Exercise 245
Section 7.1.4: Binary Decision Diagrams Exercise 245. ▶ [ M22 ] [M22] Show that the prime clauses of a monotone function f are PI(f)♯. Verified: no Solve time: 3m57s Let $f:{0,1}^n\to{0,1}$ be monotone. For $A\subseteq[n]$, define the standard assignment $x^A\in{0,1}^n$ by $$ (x^A)_i = \begin{cases} 1 & i\in A\ 0 & i\notin A. \end{cases} $$ A conjunction $t_A=\bigwedge_{i\in A} x_i$ is an implicant of $f$ iff $f(x^A)=1$, and it is...
TAOCP 7.1.4 Exercise 243
Section 7.1.4: Binary Decision Diagrams Exercise 243. [ M23 ] [M23] The closure f ∩of a family f of sets is the family of all sets that can be obtained by intersecting one or more members of f. a) Prove that f ∩= {α | α= {β | β ∈f and β ⊇α}}. b) What’s a good way to compute the ZDD for f ∩, given the ZDD for f?...
TAOCP 7.1.4 Exercise 24
Section 7.1.4: Binary Decision Diagrams Exercise 24. [ M22 ] [M22] Suppose twelve weights (w12, w13, . . . , w89) have been assigned to the edges of the grid in Fig. 22. Explain how to find a minimum spanning tree in that graph (namely, a spanning tree whose edges have minimum total weight), by applying Algo- rithm B to the BDD shown there. Verified: no Solve time: 6m43s Solution...
TAOCP 7.1.4 Exercise 236
Section 7.1.4: Binary Decision Diagrams Exercise 236. ▶ [ M25 ] [M25] Many extensions to the family algebra of exercise 203 suggest themselves when ZDDs are applied to combinatorial problems, including the following five opera- tions on families of sets: • The maximal elements f ↑= {α ∈f | β ∈f and α ⊆β implies α = β}; • The minimal elements f ↓= {α ∈f | β ∈f and...
TAOCP 7.1.4 Exercise 23
Section 7.1.4: Binary Decision Diagrams Exercise 23. ▶ [ M20 ] [M20] All paths from the root of the BDD in Fig. 22 to ⊤have exactly eight solid arcs. Why is this not a coincidence? Verified: yes Solve time: 3m14s The key point is that the number “eight” is not a property of individual paths, but a property of the construction that produced the BDD in Fig. 22. In that...
TAOCP 7.1.4 Exercise 22
Section 7.1.4: Binary Decision Diagrams Exercise 22. ▶ [ M21 ] [M21] Suppose step B3 has been simplified so that ‘Wv+1−Wvl’ and ‘Wv+1−Wvh’ are eliminated from the formulas. Prove that the algorithm will still work, when applied to BDDs that represent kernels of graphs. Verified: no Solve time: 3m50s The core issue in the previous solution is that it tried to justify the simplification by claiming a strong structural symmetry...
TAOCP 7.1.4 Exercise 211
Section 7.1.4: Binary Decision Diagrams Exercise 211. [ M20 ] [M20] When f describes the solutions to an exact cover problem, is Z(f)≤B(f)? Verified: yes Solve time: 1m57s Solution Let $f$ be the Boolean function that represents solutions of an exact cover instance on a universe $U$ with a family of subsets encoded by variables $x_1,\dots,x_n$. A solution is a choice of subsets that covers every element of $U$ exactly...
TAOCP 7.1.4 Exercise 209
Section 7.1.4: Binary Decision Diagrams Exercise 209. [ M21 ] [M21] Explain how to compute the fully elaborated truth table of a Boolean function from its ZDD representation. (See exercise 31.) Verified: no Solve time: 1m44s Let the Boolean function be given by a ZDD with variable order $x_1,x_2,\ldots,x_n$. We compute its fully elaborated truth table in lexicographic order $$ (0,\ldots,0),\ (0,\ldots,0,1),\ \ldots,\ (1,\ldots,1), $$ by recursively expanding the ZDD...
TAOCP 7.1.4 Exercise 207
Section 7.1.4: Binary Decision Diagrams Exercise 207. ▶ [ M25 ] [M25] When one or more projection functions xj are needed in applications, as in exercise 201, the following “symmetrizing” operation turns out to be very handy: (ei1 ∪ei2 ∪· · · ∪eil) § k = Sk(xi1, xi2, . . . , xil), integer k ≥0. For example, ej § 1 = xj; ej § 0 = ¯xj; (ei ∪ej)...
TAOCP 7.1.4 Exercise 205
Section 7.1.4: Binary Decision Diagrams Exercise 205. [ M25 ] [M25] Implement the five basic operations of family algebra, namely (a) f ⊔g, (b) f ⊓g, (c) f g, (d) f/g, and (e) f mod g, using the conventions of exercise 198. 274 COMBINATORIAL SEARCHING 7.1.4 Verified: no Solve time: 5m27s Solution We represent a family $f$ as a reduced ordered decision diagram over variables $x_1,x_2,\dots,x_n$, using the conventions of...
TAOCP 7.1.4 Exercise 204
Section 7.1.4: Binary Decision Diagrams Exercise 204. ▶ [ M25 ] [M25] Continuing exercise 203, two further operations are also important: • the quotient f/g = {α | α ∪β ∈f and α ∩β = ∅, for all β ∈g}. • the remainder f mod g = f \ (g ⊔(f/g)). The quotient is sometimes also called the “cofactor” of f with respect to g. a) Prove that f/(g ∪h)...
TAOCP 7.1.4 Exercise 203
Section 7.1.4: Binary Decision Diagrams Exercise 203. ▶ [ M24 ] [M24] (Family algebra.) The following algebraic conventions are useful for deal- ing with finite families of finite subsets of positive integers, and with their representation as ZDDs. The simplest such families are the empty family, denoted by ∅and represented by ⊥; the unit family {∅}, denoted by ϵ and represented by ⊤; and the elementary families {{j}} for j...
TAOCP 7.1.4 Exercise 196
Section 7.1.4: Binary Decision Diagrams Exercise 196. [ M21 ] [M21] Find a function f(x1, . . . , xn) for which Z(f) = O(n) and Z( ¯f) = Ω(n2). Verified: no Solve time: 1m42s Solution Let variables $x_1,\dots,x_n$ be interpreted as characteristic bits of a subset $S \subseteq {1,\dots,n}$, where $x_i=1$ means $i \in S$. In a zero-suppressed decision diagram (ZDD), the size $Z(f)$ counts nodes in the reduced...
TAOCP 7.1.4 Exercise 194
Section 7.1.4: Binary Decision Diagrams Exercise 194. [ M25 ] [M25] How many f(x1, . . . , xn) have the z-profile (1, . . . , 1)? (See exercise 174.) Verified: no Solve time: 4m06s Solution Let $f(x_1,\ldots,x_n)$ have truth table $\tau$, and let $f^Z$ have truth table $\tau^Z$. For $0 \le k \le n$, let $S_k(x_1,\ldots,x_n)$ denote the subfunction obtained by fixing $x_1=\cdots=x_k=1$, so its truth table is...
TAOCP 7.1.4 Exercise 193
Section 7.1.4: Binary Decision Diagrams Exercise 193. [ M21 ] [M21] Continuing exercise 192, what is SZ k (x1, . . . , xn) when 0 ≤k ≤n? Verified: no Solve time: 3m51s Solution Let $f(x_1,\ldots,x_n)$ have truth table $\tau$, and let $f^Z$ have truth table $\tau^Z$. For $0 \le k \le n$, let $S_k(x_1,\ldots,x_n)$ denote the subfunction obtained by fixing $x_1=\cdots=x_k=1$, so its truth table is the subtable of...
TAOCP 7.1.4 Exercise 192
Section 7.1.4: Binary Decision Diagrams Exercise 192. [ M20 ] [M20] Define the Z-transform of binary strings as follows: ϵZ = ϵ, 0Z = 0, 1Z = 1, and (αβ)Z = αZαZ, if |α| = n and β = 0n; αZ0n, if |α| = n and β = α; αZβZ, if |α| = |β| −1, or if |α| = |β| = n and α ̸= β ̸=...
TAOCP 7.1.4 Exercise 185
Section 7.1.4: Binary Decision Diagrams Exercise 185. [ M25 ] [M25] How large can Z(f) be, when f is a symmetric Boolean function of n variables? (See exercise 44.) Verified: no Solve time: 3m08s Solution Let $f(x_1,\dots,x_n)$ be symmetric, so its value depends only on the Hamming weight $$ w = x_1 + \cdots + x_n. $$ Hence there exist values $f_0,f_1,\dots,f_n \in {0,1}$ such that $f(x)=f_w$ whenever $|x|=w$. We...
TAOCP 7.1.4 Exercise 184
Section 7.1.4: Binary Decision Diagrams Exercise 184. [ M23 ] [M23] What are the BDD and ZDD profiles of the permutation function Pm? Verified: no Solve time: 2m53s Solution Let $P_m$ denote the Boolean predicate that encodes whether a length-$m$ assignment represents a valid permutation of ${1,\dots,m}$. In a standard encoding, each position $i \in {1,\dots,m}$ selects exactly one value in ${1,\dots,m}$, and validity means no value is used twice...
TAOCP 7.1.4 Exercise 183
Section 7.1.4: Binary Decision Diagrams Exercise 183. ▶ [ M25 ] [M25] Draw the first few levels of the BDD for the “limiting leading bit function” [(.x1x3x5 . . . )2 · (.x2x4x6 . . . )2 ≥1 2 ], which has infinitely many Boolean variables. How many nodes bk are there on level k? (We don’t allow (.x1x3x5 . . . )2 or (.x2x4x6 . . . )2 to...
TAOCP 7.1.4 Exercise 181
Section 7.1.4: Binary Decision Diagrams Exercise 181. [ M21 ] [M21] Let Lm,n be the leading bit function Z(m+n) m,n (x1, . . . , xm; y1, . . . , yn). Prove that Bmin(Lm,n) = O(2mn) when m ≤n. Verified: no Solve time: 4m53s We restart from the correct structural model of the function and rebuild the argument in a way that does not rely on an incorrect “single-bit...
TAOCP 7.1.4 Exercise 178
Section 7.1.4: Binary Decision Diagrams Exercise 178. [ M24 ] [M24] (Amano and Maruoka.) Improve the constant in (121) by using a better variable ordering: Zn(x2n−1, x1, x3, . . . , x2n−3; x2n, x2, x4, . . . , x2n−2). 272 COMBINATORIAL SEARCHING 7.1.4 Verified: no Solve time: 6m I can’t write a correct Codeforces 104120F editorial as requested because the problem statement is missing entirely (the “Problem Statement...
TAOCP 7.1.4 Exercise 177
Section 7.1.4: Binary Decision Diagrams Exercise 177. [ M22 ] [M22] Complete the proof of Theorem A by bounding the entire quasi-profile. Verified: no Solve time: 3m31s Solution The statement refers to “Theorem A” and to a “quasi-profile,” but neither is defined in the provided section excerpt. The preceding material in Section 7.1.4 up to Exercises 174–176 introduces BDD size via bead counts and a node-by-level profile $(b_0,\dots,b_n)$, but no...
TAOCP 7.1.4 Exercise 170
Section 7.1.4: Binary Decision Diagrams Exercise 170. ▶ [ M25 ] [M25] Let’s say that a Boolean function is “skinny” if its BDD involves all the variables in the simplest possible way: A skinny BDD has exactly one branch node j j for each variable xj, and either LO or HI is a sink node at every branch. a) How many Boolean functions f(x1, . . . , xn) are...
TAOCP 7.1.4 Exercise 165
Section 7.1.4: Binary Decision Diagrams Exercise 165. [ M21 ] [M21] Verify the Fibonacci-based formulas (115) for B(um) and B(vm). Verified: no Solve time: 4m52s We restart the argument from the ROBDD construction rules and avoid manipulating the coupled recurrences in an unjustified way. The key correction is that we never attempt to eliminate $B(v_m)$ algebraically; instead we track a slightly richer structural invariant that separates the two families cleanly....
TAOCP 7.1.4 Exercise 158
Section 7.1.4: Binary Decision Diagrams Exercise 158. [ M24 ] [M24] When a Boolean function f(x1, . . . , xn) is symmetrical in the variables {x1, . . . , xp}, it’s natural to expect that those variables will appear consecutively in at least one of the reorderings f π(x1, . . . , xn) that minimize B(f π). Show, however, that if f(x1, . . . , xn)...
TAOCP 7.1.4 Exercise 157
Section 7.1.4: Binary Decision Diagrams Exercise 157. [ M24 ] [M24] Show that if the m + 2m variables of the 2m-way multiplexer Mm are ar- ranged in any order such that B(M π m) > 2m+1+1, then sifting will reduce the BDD size. Verified: no Solve time: 3m42s The key failure in the previous argument is the unproven monotonicity claim: it is not true in general that swapping an...
TAOCP 7.1.4 Exercise 15
Section 7.1.4: Binary Decision Diagrams Exercise 15. [ M23 ] [M23] How many (a) independent sets and (b) kernels does Cn have, when n ≥3? Verified: no Solve time: 5m50s Let $C_n$ have vertices $1,2,\dots,n$ with edges $i\sim i\pm1 \pmod n$. A kernel is an independent dominating set. Equivalently, it is a maximal independent set: independence means no edge inside the set, and domination is exactly the maximality condition in...
TAOCP 7.1.4 Exercise 149
Section 7.1.4: Binary Decision Diagrams Exercise 149. [ M20 ] [M20] (Bollig, Löbbing, and Wegener.) Show that, in addition to Theorem J−, we also have B(f π 1 , . . . , f π m) ≤(2k −2)b0 + B(f1, . . . , fm) after a jump-down operation of k −1 levels, when (b0, . . . , bn) is the profile of {f1, . . . , fm}....
TAOCP 7.1.4 Exercise 148
Section 7.1.4: Binary Decision Diagrams Exercise 148. [ M21 ] [M21] True or false: B(f π 1 , . . . , f π m) ≤2B(f1, . . . , fm), after swapping j 1 ↔ j 2 . Verified: no Solve time: 3m55s The reviewer is correct: the statement is false , so the original proof attempt cannot be repaired. The error is not local but structural. OBDD size...
TAOCP 7.1.4 Exercise 146
Section 7.1.4: Binary Decision Diagrams Exercise 146. ▶ [ M22 ] [M22] Let (b0, . . . , bn) and (q0, . . . , qn) be a BDD base profile and quasi-profile. a) Prove that b0 ≤min(q0, (b1 + q2)(b1 + q2 −1)), b1 ≤min(b0 + q0, q2(q2 −1)), and b0 + b1 ≥q0 −q2. b) Conversely, if b0, b1, q0, and q2 are nonnegative integers that satisfy those...
TAOCP 7.1.4 Exercise 14
Section 7.1.4: Binary Decision Diagrams Exercise 14. [ M24 ] [M24] How big, exactly, are the BDDs for (a) all independent sets of the cycle graph Cn, and (b) all kernels of Cn, when n ≥3? (Number the vertices as in (12).) Verified: no Solve time: 2m20s Solution Number the vertices of the cycle $C_n$ as in the statement, so edges are ${x_i,x_{i+1}}$ for $1 \le i < n$ and...
TAOCP 7.1.4 Exercise 129
Section 7.1.4: Binary Decision Diagrams Exercise 129. [ M25 ] [M25] Define Qm(x1, . . . , xm2) to be 1 if and only if the 0–1 matrix (x(i−1)m+j) has no all-zero row and no all-zero column. Prove that B(Qπ m) = Ω(2m/m2) for all π. Verified: no Solve time: 4m04s Solution Let $S={1,\dots,m}$ denote the selector variables and $T={m+1,\dots,m+2^m}$ the data variables of the multiplexer $M_m$. For each $i\in...
TAOCP 7.1.4 Exercise 123
Section 7.1.4: Binary Decision Diagrams Exercise 123. [ M20 ] [M20] Prove formula (97), which enumerates all slates of offset s. Verified: no Solve time: 4m08s Solution Let a slate of offset $s$ be defined as in the construction preceding formula (97), where each slate is determined by a choice of $s$ distinguished positions among $n$ ordered positions, and offset means that the distinguished positions are recorded relative to a...
TAOCP 7.1.4 Exercise 121
Section 7.1.4: Binary Decision Diagrams Exercise 121. ▶ [ M22 ] [M22] If f(x1, . . . , xn) is any Boolean function, its dual f D is ¯f(¯x1, . . . , ¯xn), and its reflection f R is f(xn . . . , x1). Notice that f DD = f RR = f and f DR = f RD. a) Show that hDR n (x1, . . ....
TAOCP 7.1.4 Exercise 12
Section 7.1.4: Binary Decision Diagrams Exercise 12. ▶ [ M21 ] [M21] A kernel of a digraph is a set of vertices K such that v ∈K implies v ̸−−→u for all u ∈K; v /∈K implies v −−→u for some u ∈K. a) Show that when the digraph is an ordinary graph (that is, when u−−→v if and only if v −−→u), a kernel is the same as a...
TAOCP 7.1.4 Exercise 118
Section 7.1.4: Binary Decision Diagrams Exercise 118. [ M23 ] [M23] Show that, from the perspective of Section 7.1.2, the hidden weighted bit function has cost C(hn) = O(n). What is the exact value of C(h4)? Verified: no Solve time: 3m18s Solution Let $g(k)$ be the Gray binary code defined in (7), equivalently $g(k)=k\oplus \lfloor k/2\rfloor$ by (9). Let $k,l$ satisfy $0\le k,l<2^n$, and let $g(k)$ and $g(l)$ differ in...
TAOCP 7.1.4 Exercise 117
Section 7.1.4: Binary Decision Diagrams Exercise 117. [ M20 ] [M20] Compare Q(f) to B(f) when f = Mm(x1, . . . , xm; xm+1, . . . , xm+2m). Verified: no Solve time: 4m59s Solution Let $f = M_m(x_1,\ldots,x_m; x_{m+1},\ldots,x_{2m})$, where $M_m$ denotes the equality function on two $m$-bit blocks, so that $f=1$ if and only if $x_i = x_{m+i}$ for all $1 \le i \le m$. For a...
TAOCP 7.1.4 Exercise 116
Section 7.1.4: Binary Decision Diagrams Exercise 116. [ M21 ] [M21] What is the (a) worst case (b) average case of a random quasi-profile? Verified: no Solve time: 4m27s Solution Let the quasi-profile of a BDD for a Boolean function $f(x_1,\dots,x_n)$ be the sequence $Q_k(f)$, where $Q_k(f)$ counts the number of distinct nodes (equivalently distinct subfunctions) at level $k$, corresponding to restrictions $f(x_1,\dots,x_k,;x_{k+1},\dots,x_n)$ as in the bead construction of Section...
TAOCP 7.1.4 Exercise 115
Section 7.1.4: Binary Decision Diagrams Exercise 115. ▶ [ M22 ] [M22] Prove the quasi-profile inequalities (84) and (124). Verified: no Solve time: 3m15s Solution Let $p_k$ denote the number of beads (BDD nodes) at level $k$, and let $q_k$ denote the number of distinct subtables produced at level $k$ before reduction, in the sense of Section 7.1.4. A subfunction at level $k$ arises from fixing $x_1,\dots,x_{k-1}$ and considering the...
TAOCP 7.1.4 Exercise 111
Section 7.1.4: Binary Decision Diagrams Exercise 111. [ M22 ] [M22] Verify the summation formula (79) in Theorem U. Verified: no Solve time: 7m07s Solution Let $\Gamma_3 = (g(0), g(1), \dots, g(7))$ be the 3-bit Gray binary code in cyclic order, so consecutive terms including $g(7) \to g(0)$ differ in exactly one bit, by the defining property of Gray code in Section 7.2.1.1. Define a bijection from the $8 \times...
TAOCP 7.1.3 Exercise 42
Section 7.1.3: Bitwise Tricks and Techniques Exercise 42. [ M21 ] If $u = 2^{e_1} + \cdots + 2^{e_r}$, with $e_1 > \cdots > e_r \ge 0$, express the sum $\sum_{k=0}^{r-1} \nu k$ in terms of the exponents $e_1, \ldots, e_r$. Verified: yes Solve time: 4m06s Solution Let $u = 2^{e_1} + \cdots + 2^{e_r}$ with $e_1 > \cdots > e_r \ge 0$. Let $\nu x$ denote the number of...
TAOCP 7.1.1 Exercise 129
Section 7.1.1: Boolean Basics Exercise 129. [ M23 ] How many different canalizing functions of $n$ variables exist? Verified: no Solve time: 4m26s Solution Let $C_n$ denote the number of canalizing Boolean functions on ${0,1}^n$. For $1\le i\le n$, $\alpha,\beta\in{0,1}$, let $$ A_{i,\alpha,\beta} {,f:{0,1}^n\to{0,1} \mid x_i=\alpha \Longrightarrow f=\beta ,}. $$ A function is canalizing if and only if it belongs to at least one of the $4n$ sets $A_{i,\alpha,\beta}$. Thus...
TAOCP 4.3.3 Exercise 13
Section 4.3.3: How Fast Can We Multiply? Exercise 13. [ M25 ] What is a good upper bound on the time needed to multiply an $m$-bit number by an $n$-bit number, when both $m$ and $n$ are very large but $n$ is much larger than $m$, based on the results discussed in this section for the case $m = n$? Verified: yes Solve time: 1m52s Solution Let $M$ be an...
TAOCP 3.2.1.2 Exercise 13
Section 3.2.1.2: Choice of Multiplier Exercise 13. [ M22 ] [M22] Let $p$ be prime. Given that $a$ is not a primitive element modulo $p$, show that either $a$ is a multiple of $p$ or $a^{(p-1)/q} \equiv 1 \pmod{p}$ for some prime number $q$ that divides $p - 1$. Verified: no Solve time: 1m18s Solution Let $p$ be prime and suppose $a$ is not a primitive element modulo $p$. The...
TAOCP 3.2.1.1 Exercise 9
Section 3.2.1.1: Choice of Modulus Exercise 9. ▶ [ M25 ] Most high-level programming languages do not provide a good way to divide a two-word integer by a one-word integer, nor do they provide the hmuIt operation of exercise 3. The purpose of this exercise is to find a reasonable way to cope with such limitations when we wish to evaluate $ax \bmod m$ for variable $x$ and for constants...
TAOCP 4.7 Exercise 17
Section 4.7: Manipulation of Power Series Exercise 17. ▶ [ M20 ] Given the power series $V(z) = V_1 z + V_2 z^2 + V_3 z^3 + \cdots$, we define the power matrix of $V$ as the infinite array of coefficients $v_{nk} = \frac{1}{n} [z^k] V(z)^n$; the $n$th poweroid of $V$ is then defined to be $V_n(x) = v_{n0} + v_{n1} x + \cdots + v_{nn} x^n$. Prove that poweroids...
TAOCP 3.4.2 Exercise 17
Section 3.4.2: Random Sampling and Shuffling Exercise 17. [ M22 ] (R. W. Floyd.) Prove that the following algorithm generates a random sample $S$ of $n$ integers from ${1, \ldots, N}$: Set $S \leftarrow \emptyset$; then for $j \leftarrow N - n + 1,\ N - n + 2,\ \ldots,\ N$ (in this order), set $k \leftarrow \lfloor jU \rfloor + 1$ and $$S \leftarrow \begin{cases} S \cup {k}, &...
TAOCP 4.5.2 Exercise 35
Section 4.5.2: The Greatest Common Divisor Exercise 35. [ M25 ] [M25] Consider a directed graph with vertices $(m, n)$ for all nonnegative integers $m, n \ge 0$, having arcs from $(m, n)$ to $(m', n')$ whenever it is possible for a subtract-and-shift cycle of Algorithm B to transform integers $u$ and $v$ with $\lfloor \lg u \rfloor = m$ and $\lfloor \lg v \rfloor = n$ into integers $u'$...
TAOCP 4.6.1 Exercise 22
Section 4.6.1: Division of Polynomials Exercise 22. [ M23 ] Prove Sturm's theorem. [ Hint: Some sign sequences are impossible.] Verified: no Solve time: 3m27s Let $f_0=f,\ f_1=f',\ f_{i+1}=-\operatorname{rem}(f_{i-1},f_i)$ be the Sturm sequence, and let the process terminate with a nonzero constant $f_s$. Define $V(x)$ for real $x$ not a root of any $f_i$ as the number of sign changes in $$ (f_0(x),f_1(x),\dots,f_s(x)) $$ after deleting zeros. We prove that...
TAOCP 4.6.1 Exercise 18
Section 4.6.1: Division of Polynomials Exercise 18. ▶ [ M24 ] $[M24]$ (Euclidean algorithm for string polynomials.) Let $V_1$ and $V_2$ be string polynomials, not both zero, having a common left multiple . (This means that there exist string polynomials $U_1$ and $U_2$, not both zero, such that $U_1V_1 = U_2V_2$.) The purpose of this exercise is to find an algorithm to compute their greatest common right divisor $\gcd(V_1, V_2)$...
TAOCP 3.4.1 Exercise 14
Section 3.4.1: Numerical Distributions Exercise 14. [ M21 ] If $X$ is a random variable with the continuous distribution $F(x)$, and if $c$ is a (possibly negative) constant, what is the distribution of $cX$? Verified: yes Solve time: 1m36s Solution Let $X$ have continuous distribution function $F(x)=\Pr{X\le x}$ and define $Y=cX$, where $c$ is a constant. The distribution function of $Y$ is $G(y)=\Pr{Y\le y}=\Pr{cX\le y}.$ Three cases determine how the...
TAOCP 4.3.1 Exercise 4
Section 4.3.1: The Classical Algorithms Exercise 4. [ M21 ] Give a formal proof of the validity of Algorithm A, using the method of inductive assertions explained in Section 1.2.1. Verified: no Solve time: 1m30s Solution Let $U = (u_{n-1}\ldots u_0) b,\quad V = (v {n-1}\ldots v_0) b,\quad W = (w {n-1}\ldots w_0)_b.$ Define for each $j$ with $0 \le j \le n$ the partial values $U_j = \sum_{i=0}^{j-1} u_i...
TAOCP 3.2.2 Exercise 34
Section 3.2.2: Other Methods Exercise 34. [ M25 ] Prove that the inversive congruential sequence (12) has period $p + 1$ if and only if the polynomial $f(x) = x^2 - cx - a$ has the following two properties: (i) $x^{p+1} \bmod f(x)$ is a nonzero constant, when computed with polynomial arithmetic modulo $p$; (ii) $x^{(p+1)/q} \bmod f(x)$ has degree 1 for every prime $q$ that divides $p+1$. [ Hint:...
TAOCP 1.3.3 Exercise 28
Section 1.3.3: Applications to Permutations Exercise 28. [ M21 ] (I. Kaplansky.) If the "Josephus permutation" defined in exercise 1.3.2-22 is expressed in cycle form, we obtain $(1,5,3,6,8,2,4)(7)$ when $n = 8$ and $m = 4$. Show that this permutation in the general case is the product $(n\ n!-!1\ \ldots\ 2\ 1)^{m-1} \times (n\ n!-!1\ \ldots\ 2)^{m-1}\cdots (n\ n!-!1)^{m-1}$. Verified: no Solve time: 4m28s Let the Josephus process on ${1,\dots,n}$...
TAOCP 1.3.3 Exercise 25
Section 1.3.3: Applications to Permutations Exercise 25. [ M22 ] Prove Eq. (29). Verified: no Solve time: 4m30s Let $$ T(n)=\sum_{k=1}^{n}(k-1)(n-k)!. $$ We derive Eq. (29) directly from this definition, without introducing unproved structural decompositions. 1. Change of index Let $j=n-k$. Then $k=n-j$, and as $k$ runs from $1$ to $n$, $j$ runs from $n-1$ down to $0$. Hence $$ T(n)=\sum_{j=0}^{n-1}(n-j-1),j!. $$ Split the factor: $$ T(n)=(n-1)\sum_{j=0}^{n-1}j!-\sum_{j=0}^{n-1}j\cdot j!. $$ 2....
TAOCP 1.3.3 Exercise 20
Section 1.3.3: Applications to Permutations Exercise 20. [ M20 ] Given that all singleton cycles are written out explicitly, how many different ways are there to write the cycle notation of a permutation that has $\alpha_1$ one-cycles, $\alpha_2$ two-cycles, ... ? (See exercise 5.) Verified: yes Solve time: 2m09s Solution Let a permutation of $n$ elements have exactly $\alpha_j$ cycles of length $j$, for $1 \le j \le n$, including...
TAOCP 1.3.3 Exercise 13
Section 1.3.3: Applications to Permutations Exercise 13. [ M24 ] Prove that Algorithm $J$ is valid. Verified: no Solve time: 2m59s The flaw in the previous response is that it refused the task instead of engaging with the standard definition of Algorithm $J$ in TAOCP §1.3.3. In that section, Algorithm $J$ is the Johnson–Trotter permutation generation algorithm. Its validity means: it produces each permutation of ${1,\dots,n}$ exactly once and then...
TAOCP 4.6 Exercise 5
Section 4.6: Polynomial Arithmetic Exercise 5. ▶ [ M21 ] Show how to multiply two polynomials of degree $\le n$, modulo 2, with an execution time proportional to $O(n^{1.5})$ when $n$ is large, by adapting Karatsuba's method (see Section 4.3.3). It is possible to divide one polynomial by another in essentially the same way that we divide one multiple-precision integer by another, when arithmetic is being done on polynomials over...
TAOCP 4.6 Exercise 3
Section 4.6: Polynomial Arithmetic Exercise 3. [ M20 ] If each of the coefficients $u_s, \ldots, u_0, v_s, \ldots, v_0$ in (4) is an integer satisfying the conditions $|u_i| \le m_1$, $|v_j| \le m_2$, what is the maximum absolute value of the product coefficients $w_k$? Verified: yes Solve time: 1m22s Solution From (4), $$ w_k = u_0 v_k + u_1 v_{k-1} + \cdots + u_k v_0, $$ where terms with...
TAOCP 4.2.4 Exercise 13
Section 4.2.4: Distribution of Floating Point Numbers Exercise 13. ▶ [ M20 ] The floating point multiplication routine, Algorithm 4.2.1M, requires zero or one left shifts during normalization, depending on whether $f_u f_v \ge 1/b$ or not. Assuming that the input operands are independently distributed according to the logarithmic law, what is the probability that no left shift is needed for normalization of the result? Verified: yes Solve time: 2m...
TAOCP 4.3.1 Exercise 17
Section 4.3.1: The Classical Algorithms Exercise 17. [ M20 ] In the notation of Fig. 6, assume that $v_{n-1} \ge \lfloor b/2 \rfloor$; show that if $u_n = v_{n-1}$, we must have $q = b - 1$ or $b - 2$. Verified: no Solve time: 4m10s Let $V = (v_{n-1}\ldots v_0) b$ be the divisor and let $R$ be the partial remainder at the moment the quotient digit $q$ is...
TAOCP 6.5 Exercise 19
Section 6.5: Retrieval on Secondary Keys Exercise 19. [ M22 ] [M22] Analyze the ABD(8,5) of (15), as (13) has been analyzed in (14): How many of the 32 locations must be searched for an average query with k bits unspecified? How many must be searched in the worst case? Verified: yes Solve time: 3m54s The error in the previous solution comes from treating ABD(8,5) as if query elements were...
TAOCP 6.5 Exercise 2
Section 6.5: Retrieval on Secondary Keys Exercise 2. [ M25 ] [M25] (Sakti P. Ghosh.) Find the minimum possible length / of a list rir2...11 of references to records, such that the set of all responses to any of the inclusive queries 4 1, 1 , Lee, #11, 1 1, 11*, 111 on three binary-valued secondary keys will appear in consecutive locations r;...1;. Verified: no Solve time: 5m25s Let the...
TAOCP 6.5 Exercise 11
Section 6.5: Retrieval on Secondary Keys Exercise 11. [ M25 ] [M25] The text describes the use of Steiner triple systems in connection with inclusive queries; in order to extend this to all basic queries it is natural to define the following concept. A complemented triple system of order v is an arrangement of 2vu objects {11,...,2v,%1,...,Zv} into triples such that every pair of objects occurs together in exactly one...
TAOCP 6.5 Exercise 9
Section 6.5: Retrieval on Secondary Keys Exercise 9. [ M20 ] [M20] Prove that when v = 3”, the set of all triples of the form {(a1 «++ Qk-1 0 by wae bn, k)3, (ai ++» QAk-1 1 C1... Cn, k)3, (a1 ~.-Ak-1 2d, wae dn, k)3}, 1<k<vn, forms a Steiner triple system, where the a’s, b’s, c’s, and d’s range over all combinations of 0s, 1s, and 2s such...
TAOCP 6.5 Exercise 7
Section 6.5: Retrieval on Secondary Keys Exercise 7. [ M24 ] [M24] (R.L. Rivest.) Find the functions L(t), as defined in the previous exercise, for the following combinatorial hash functions: (a) m=3,n=2 (b) m=4,n=2 00*-0 00*« «-0 1 O0>1 1l x0751 *1l1l, -2 lll, -2 101-3 101l , 2 010-3 *1l01>3 100*, +3 Verified: no Solve time: 3m59s The fundamental issue in the proposed solution is not computational but logical:...
TAOCP 6.5 Exercise 13
Section 6.5: Retrieval on Secondary Keys Exercise 13. [ M25 ] [M25] Construct quadruple systems with v = 4” elements, analogous to the triple system of exercise 9. Verified: yes Solve time: 2m25s Let $m = 2n$ and let $V = \mathbb{F}_2^m$, so $|V| = 2^m = 4^n$. We construct blocks that are exactly the affine 2-dimensional subspaces of $V$. Construction For any $x \in V$ and any linearly independent...
TAOCP 5.4.2 Exercise 11
Section 5.4.2: The Polyphase Merge Exercise 11. [ M24 ] [M24] Prove that the nth element of the string Qo. in (12) is equal to the number of distinct Fibonacci numbers in the fifth-order Fibonacci representation of n, 1. [See exercise 10.] 010 0 0 0010 0 Verified: no Solve time: 4m A clean proof must eliminate the earlier two failures: (i) treating both objects as sharing an unproved “common...
TAOCP 5.3.1 Exercise 8
Section 5.3.1: Minimum-Comparison Sorting Exercise 8. [ M23 ] [M23] (Optimum exchange sorting.) Every exchange sorting algorithm as defined in Section 5.2.2 can be represented as a comparison-exchange tree, namely a binary tree structure whose internal nodes have the form $i : j$ for $i < j$, interpreted as the following operation: “If $K_i < K_j$, continue by taking the left branch of the tree; if $K_i > K_j$, continue...
TAOCP 6.3 Exercise 36
Section 6.3: Digital Searching Exercise 36. [ M25 ] [M25] There are five binary trees with three internal nodes. If we consider how frequently each particular one of these occurs as the search tree in various algorithms, for random data, we find the following different probabilities: ar Tree search 1 1 1 1 1 (Algorithm 6.2.2T) 6 6 3 6 6 Digital tree search 1 1 1 1 1 (Algorithm...
TAOCP 5.4.8 Exercise 4
Section 5.4.8: Two-Tape Sorting Exercise 4. [ M23 ] [M23] In the elevator problem, suppose there are indices p and q, with gq > p+2, Up > 0, Ug > 0, and up41 = ++: = Ug-1 = 0. Explain how to construct a schedule requiring at most (9) units of time. [M23] True or false: After step K1 of the algorithm in Theorem K, nobody on the elevator has...
TAOCP 6.2.2 Exercise 19
Section 6.2.2: Binary Tree Searching Exercise 19. [ M23 ] [M23] What is the approximate average number of comparisons, (11), when the input probabilities satisfy the 80-20 law defined in Eq. 6.1, (11)? Verified: yes Solve time: 1m23s Solution Let $N$ keys be inserted in random order into a binary search tree generated by Algorithm T. For a successful search, the number of comparisons for a key is $d(x)+1$, where...
TAOCP 6.1 Exercise 7
Section 6.1: Sequential Searching Exercise 7. [ M20 ] [M20] Evaluate the average number of comparisons, (3), using the “binary” probability distribution (5). Verified: yes Solve time: 1m01s Solution The probability distribution (5) is $$ p_k = 2^{-k}\quad (1 \le k \le N-1), \qquad p_N = 2^{-(N-1)}. $$ The average number of comparisons in a successful search is given by (3): $$ C_N = \sum_{k=1}^{N} k p_k = \sum_{k=1}^{N-1} \frac{k}{2^k}...
TAOCP 5.4.2 Exercise 8
Section 5.4.2: The Polyphase Merge Exercise 8. [ M20 ] [M20] (E. Netto, 1901.) Let N® be the number of ways to express m as an ordered sum of the integers {1,2,...,p}. For example, when p = 3 and m = 5, there are 13 ways, namely 1+1+1+1+1 = 1414142 = 1414241 =14143=1+42+141= 14242=14341=2414141=24142=24241=243=34141=342. Show that N®) is a generalized Fibonacci number. Verified: yes Solve time: 1m29s Solution Let $N_m^{(p)}$...
TAOCP 6.3 Exercise 37
Section 6.3: Digital Searching Exercise 37. [ M22 ] [M22] Consider a binary tree with b; external nodes on level J. The text observes that the running time for unsuccessful searching in digital search trees is not directly related to the external path length 5~*/b;, but instead it is essentially proportional to the modified external path length S 1b:2 '. Prove or disprove: The smallest modified external path length, over...
TAOCP 6.1 Exercise 17
Section 6.1: Sequential Searching Exercise 17. [ M23 ] [M23] (J. R. Jackson.) Suppose you have to do n jobs; the jth job takes T; units of time, and it has a deadline D;. In other words, the jth job is supposed to be finished after at most D; units of time have elapsed. What schedule a1 a2...an for processing the jobs will minimize the maximum tardiness, namely max(Ty,, Da,,...
TAOCP 6.2.2 Exercise 21
Section 6.2.2: Binary Tree Searching Exercise 21. [ M20 ] [M20] If p, gq, r are probabilities chosen at random, subject to the condition that p+q+r =1, what are the probabilities that trees I, II, III, IV, V of (13) are optimal, respectively? (Consider the relative areas of the regions in Fig. 14.) Verified: no Solve time: 5m16s The solution fails at the very first structural step: the cost formulas...
TAOCP 6.2.1 Exercise 13
Section 6.2.1: Searching an Ordered Table Exercise 13. [ M24 ] [M24] Tabulate the average number of comparisons made by Shar’s method, for 1< N < 16, considering both successful and unsuccessful searches. Verified: no Solve time: 4m33s Shar’s method in this exercise is the standard binary search method on an ordered table. The number of comparisons equals the number of nodes visited in the implicit binary decision tree induced...
TAOCP 6.2.2 Exercise 39
Section 6.2.2: Binary Tree Searching Exercise 39. [ M25 ] [M25] Let wi, ..., wn be nonnegative weights with wi +---+ wn = 1. Prove that the weighted path length of the Huffman tree constructed in Section 2.3.4.5 is less than H(wi,...,Wn) +1. Hint: See the proof of Theorem M. Verified: yes Solve time: 1m12s Solution Let $w_1,\dots,w_n$ be nonnegative with $w_1+\cdots+w_n=1$. Let $H(w_1,\dots,w_n)= -\sum_{i=1}^n w_i \lg w_i$, where $\lg...
TAOCP 6.2.2 Exercise 38
Section 6.2.2: Binary Tree Searching Exercise 38. [ M20 ] [M20] Explain why Theorem M holds in general, although we have only proved it in the case so < 81 < 82 <++: < Sn. Verified: yes Solve time: 1m27s Solution Let $s_0, s_1, \ldots, s_n$ be arbitrary distinct keys. Let $r(s_i)$ denote the rank of $s_i$ in increasing order, so that ${r(s_0),\ldots,r(s_n)}={1,\ldots,n+1}$ and $s_i < s_j$ if and only...
TAOCP 6.2.1 Exercise 15
Section 6.2.1: Searching an Ordered Table Exercise 15. [ M19 ] [M19] For what values of k does the Fibonacci tree of order & define an optimal search procedure, in the sense that the fewest comparisons are made on the average? A424 SEARCHING 6.2.1 Verified: yes Solve time: 1m36s Solution Let $T_k$ denote the Fibonacci tree of order $k$. Let $N_k$ be its number of internal (circular) nodes. From the...
TAOCP 6.2.2 Exercise 25
Section 6.2.2: Binary Tree Searching Exercise 25. [ M20 ] [M20] Let A and B be nonempty sets of real numbers, and define A < B if the following property holds: (ae A, bE B, and b <a) implies (a € Band be A). a) Prove that this relation is transitive on nonempty sets. b) Prove or disprove: A < B if and only if A< AUB< B. Verified: no...
TAOCP 5.1.4 Exercise 10
Section 5.1.4: Tableaux and Involutions Exercise 10. [ M20 ] [M20] How many permutations of the multiset ${1^a, 2^b, 3^c}$ have the property that, as we read the permutation from left to right, the number of 3’s never exceeds the number of 2’s, and the number of 2’s never exceeds the number of 1’s? (For example, $aabcabbcac$ is such a permutation.) Verified: no Solve time: 16m42s Solution Let $$ N(a,b,c)...
TAOCP 6.3 Exercise 17
Section 6.3: Digital Searching Exercise 17. [ M25 ] [M25] Find a way to express the solution of the recurrence to = 21 = 0, tm = an tml" SO(n = 1) Fa, n > 2, in terms of binomial transforms, by generalizing the technique of exercise 5.2.2, 36. 6.3 DIGITAL SEARCHING 509 Verified: no Solve time: 4m59s Let $B$ denote the binomial transform operator acting on sequences $x =...
TAOCP 6.2.2 Exercise 22
Section 6.2.2: Binary Tree Searching Exercise 22. [ M20 ] [M20] Prove that r[i, 7-1] is never greater than r[i+1, j] when step K4 of Algorithm K is performed. Verified: no Solve time: 3m56s Let $$ e[i,j]=\min_{k=i}^j\bigl(e[i,k-1]+e[k+1,j]+w[i,j]\bigr), \qquad r[i,j]\in\arg\min. $$ Fix $i<j-1$ and set $$ a=r[i,j-1],\qquad b=r[i+1,j]. $$ We prove $$ r[i,j-1]\le r[i+1,j], \quad\text{i.e. } a\le b. $$ 1. Reduction to the decision function Define the split-cost function without the...
TAOCP 5.2.1 Exercise 13
Section 5.2.1: Sorting by Insertion Exercise 13. [ M16 ] [M16] Explain how to put weights on the horizontal line segments of a lattice, instead of the vertical segments, so that the sum of the horizontal weights on a lattice path is the number of inversions in the corresponding 2-ordered permutation. Verified: yes Solve time: 2m55s Start from the standard lattice representation of a permutation used in Section 5.2.1: a...
TAOCP 6.3 Exercise 25
Section 6.3: Digital Searching Exercise 25. [ M24 ] [M24] The mathematical derivations of asymptotic values in this section have been quite difficult, involving complex variable theory, because it is desirable to get more than just the leading term of the asymptotic behavior (and the second term is intrinsically complicated). The purpose of this exercise is to show that elementary methods are good enough to deduce some of the results...
TAOCP 6.2.1 Exercise 17
Section 6.2.1: Searching an Ordered Table Exercise 17. [ M21 ] [M21] From exercise 1.2.8-34 (or exercise 5.4.2-10) we know that every positive integer n has a unique representation as a sum of Fibonacci numbers nm = Fo, + Foo +++++ Fa,,; where r > 1, aj > aj41+2 forl1 <j <r,anda, > 2. Prove that in the Fibonacci tree of order k, the path from the root to node...
TAOCP 5.4.9 Exercise 15
Section 5.4.9: Disks and Drums Exercise 15. [ M22 ] [M22] Show that Theorem F can be improved slightly if the elevator is initially empty and if F(b)n 4 t: At least [(F(b)n + m, t)/(b+ m)] stops are necessary in such a case. Verified: no Solve time: 3m25s Solution Let the elevator process be measured in stops, and let each stop be a position at which the elevator services...
TAOCP 6.2.2 Exercise 7
Section 6.2.2: Binary Tree Searching Exercise 7. [ M25 ] [M25] (S. R. Arora and W. T. Dent.) After n elements have been inserted into an initially empty tree, in random order, what is the average number of comparisons needed by Algorithm T to find the mth largest element, given the key of that element? Verified: yes Solve time: 1m16s Solution Let keys $1,2,\dots,n$ be inserted in random order to...
TAOCP 6.2.2 Exercise 18
Section 6.2.2: Binary Tree Searching Exercise 18. [ M21 ] [M21] Show that Zipf’s law yields (12). 456 SEARCHING 6.2.2 Verified: no Solve time: 3m56s Start by separating what must be proved from what was previously assumed without justification. The exercise asks to show that Zipf’s law implies TAOCP equation (12). Therefore the first step is to identify equation (12) from §6.2.2 and connect it to the Zipf model, not...
TAOCP 6.2.2 Exercise 24
Section 6.2.2: Binary Tree Searching Exercise 24. [ M25 ] [M25] Given that pn = gn = 0 and that the other weights are nonnegative, prove that an optimum tree for (p1,...,Pn; Go,---;Qn) may be obtained by replacing by n-1 in any optimum tree for (pi,...,DPn, 1; Go,--+-,@n, 1)- Verified: no Solve time: 4m32s Let $T$ be a binary search tree with cost $$ C(T)=\sum_{i=1}^n p_i,\mathrm{depth}(k_i)+\sum_{i=0}^n q_i,\mathrm{depth}(d_i), $$ where all...
TAOCP 5.4.2 Exercise 12
Section 5.4.2: The Polyphase Merge Exercise 12. [ M18 ] [M18] Find a connection between powers of the matrix |} 0 0 0 1 O | and 00001 11111 the perfect Fibonacci distributions in (1). Verified: no Solve time: 5m The flaw in the previous argument is not the linear algebraic part but the missing derivation of the transition rule from the definition of the perfect Fibonacci (perfect polyphase) distributions...
TAOCP 6.2.2 Exercise 5
Section 6.2.2: Binary Tree Searching Exercise 5. [ M25 ] [M25] There are 11! = 39,916,800 different orders in which the names CAPRICORN, AQUARIUS, etc. could have been inserted into a binary search tree. a) How many of these arrangements will produce Fig. 10? b) How many of these arrangements will produce a degenerate tree, in which LLINK or RLINK is A in each node? Verified: yes Solve time: 1m46s...
TAOCP 5.4.9 Exercise 5
Section 5.4.9: Disks and Drums Exercise 5. [ M20 ] [M20] When two disks are being used, so that reading on one is overlapped with writing on the other, we cannot use merge patterns like that of Fig. 93 since some leaves are at even levels and some are at odd levels. Show how to modify the construction of Theorem K in order to produce trees that are optimal subject...
TAOCP 6.3 Exercise 32
Section 6.3: Digital Searching Exercise 32. [ M21 ] [M21] What is the average value of the sum of all SKIP fields in a random Patrician tree with N, 1 internal nodes? Verified: no Solve time: 1m50s Solution Let keys be infinite binary sequences generated by independent unbiased bits. The Patrician tree is the compressed binary trie in which each internal node stores a SKIP value equal to the number...
TAOCP 5.3.2 Exercise 3
Section 5.3.2: Minimum-Comparison Merging Exercise 3. [ M24 ] [M24] Prove that $\underline{M}(1,n) = M(1,n)$ for all $n$. Verified: yes Solve time: 2m21s We restart from the definitions of the two quantities in Knuth’s merging model. Let $A_1$ be a single-element sorted list and $B_1 < \cdots < B_n$ a sorted list. A merge proceeds by comparisons only between the current “heads” of the two lists. Since $A_1$ is always...
TAOCP 5.4.6 Exercise 10
Section 5.4.6: Practical Considerations for Tape Merging Exercise 10. [ M23 ] [M23] Use the analyses in Sections 5.4.2 and 5.4.3 to show that the length of each rewind during a standard six-tape polyphase or cascade merge is rarely more than about 54 percent of the file (except for the initial and final rewinds, which cover the entire file). Verified: no Solve time: 4m32s Let $S$ be the total number...
TAOCP 6.1 Exercise 1
Section 6.1: Sequential Searching Exercise 1. [ M20 ] [M20] When all the search keys are equally probable, what is the standard deviation of the number of comparisons made in a successful sequential search through a table of N records? Verified: yes Solve time: 1m01s Solution In a successful sequential search through $N$ records, every position $i \in {1,\dots,N}$ occurs with probability $1/N$. The number of comparisons is therefore the...
TAOCP 6.3 Exercise 31
Section 6.3: Digital Searching Exercise 31. [ M21 ] [M21] Solve the recurrence (16). Verified: no Solve time: 4m43s Let $A_n$ be the expected cost of an $M$-ary digital search tree built from $n$ random keys, and let $P(z)$ be its Poisson transform. 1. Poissonized recurrence As in the proposed solution, Poissonization with parameter $z$ gives independent subtree sizes $$ N_0,\dots,N_{M-1} \sim \mathrm{Poisson}(z/M), $$ and hence the exact functional equation...
TAOCP 6.2.1 Exercise 7
Section 6.2.1: Searching an Ordered Table Exercise 7. [ M22 ] [M22] Would Algorithm U still work properly if we changed step U1 so that a) both 7 and m are set equal to | N/2|? b) both i and m are set equal to [N/2]? [Hint: Suppose the first step were “Set i ~, 0, m< N (or N +1), go to U4.”| Verified: yes Solve time: 3m04s The...
TAOCP 5.4.1 Exercise 11
Section 5.4.1: Multiway Merging and Replacement Selection Exercise 11. [ M20 ] [M20] True or false: The probability that KEY(Q) < LASTKEY in step R4 is approximately 50%, assuming random input. Verified: no Solve time: 4m09s Corrected solution We analyze the random-permutation model: all $n!$ input permutations of distinct keys are equally likely. The algorithm (replacement selection) is deterministic and comparison-based, so for each input permutation it produces a well-defined...
TAOCP 6.2.1 Exercise 5
Section 6.2.1: Searching an Ordered Table Exercise 5. [ M24 ] [M24] For what values of N is Program B actually slower than a sequential search (Program 6.1Q’) on the average, assuming that the search is successful? Verified: yes Solve time: 2m29s We restart from the correct inequality and determine the full integer solution set carefully. Let $$ T_B(N) = (18 \lg N - 16)u, \qquad T_S(N) = \frac{N+1}{2}u. $$...
TAOCP 5.4.2 Exercise 22
Section 5.4.2: The Polyphase Merge Exercise 22. [ M21 ] [M21] Each term of the sequence (24) is approximately equal to the sum of the previous two. Does this phenomenon hold for the remaining numbers of the sequence? Formulate and prove a theorem about tn, tn, 1, tn, 2. Verified: no Solve time: 1m21s Solution Let $T = P+1$ and let $t_n$ denote the total number of runs in the...
TAOCP 5.2.3 Exercise 12
Section 5.2.3: Sorting by Selection Exercise 12. [ M22 ] [M22] (B. A. Chartres.) Design a high-speed algorithm that builds a table of the prime numbers < N, making use of a priority queue to avoid division operations. [Hint: Let the smallest key in the priority queue be the least odd nonprime number greater than the last odd number considered as a prime candidate. Try to minimize the number of...
TAOCP 6.1 Exercise 10
Section 6.1: Sequential Searching Exercise 10. [ M20 ] [M20] The best arrangement of records in a sequential table is specified by (4); what is the worst arrangement? Show that the average number of comparisons in the worst arrangement has a simple relation to the average number of comparisons in the best arrangement. Verified: yes Solve time: 1m09s Solution Let $p_1, p_2, \dots, p_N$ be the probabilities that the argument...
TAOCP 5.3.3 Exercise 8
Section 5.3.3: Minimum-Comparison Selection Exercise 8. [ M20 ] [M20] If groups of size 5, not 7, had been used in the proof of Theorem L, what theorem would have been obtained? Verified: yes Solve time: 2m22s Let the selection algorithm be modified so that the input is partitioned into groups of 5 instead of groups of 7, and the median-of-medians is used as the pivot exactly as in the...
TAOCP 6.2.2 Exercise 26
Section 6.2.2: Binary Tree Searching Exercise 26. [ M22 ] [M22] Let (pi,...,Pn3 Go,---;Qn) be nonnegative weights, where pn + qn = Z. Prove that as x varies from 0 to oo, while (p1,...,Dn, 1; Go,---,@n, 1) are held constant, the cost c(0,n) of an optimum binary search tree is a concave, continuous, piecewise linear function of x with integer slopes. In other words, prove that there exist positive integers...
TAOCP 5.3.1 Exercise 7
Section 5.3.1: Minimum-Comparison Sorting Exercise 7. [ M25 ] [M25] (S. M. Chase.) Let $a_1, a_2, \ldots, a_n$ be a permutation of ${1, 2, \ldots, n}$. Prove that any algorithm that decides whether this permutation is even or odd (that is, whether it has an even or odd number of inversions), based solely on comparisons between the $a$'s, must make at least $n \lg n$ comparisons, even though the algorithm...
TAOCP 5.2.4 Exercise 8
Section 5.2.4: Sorting by Merging Exercise 8. [ M24 ] [M24] Let the binary representation of $N$ be $2^{e_1} + 2^{e_2} + \dots + 2^{e_t}$, where $e_1 > e_2 > \dots > e_t > 0, t > 1$. Prove that the maximum number of key comparisons performed by Algorithm L is $$1 - 2^0 + \sum (e_i + k - 1) 2^0.$$ Verified: no Solve time: 4m11s The error in...
TAOCP 5.3.2 Exercise 12
Section 5.3.2: Minimum-Comparison Merging Exercise 12. [ M21 ] [M21] Let $R_n(i,j)$ be the least number of comparisons required to sort the distinct objects ${\alpha, \beta, X_1, \ldots, X_n}$, given the relations $$\alpha < \beta,\qquad X_1 < X_2 < \cdots < X_n,\qquad \alpha < X_{i+1},\qquad \beta > X_{n-j}.$$ (The condition $\alpha < X_{i+1}$ or $\beta > X_{n-j}$ becomes vacuous when $i \ge n$ or $j \ge n$. Therefore $R_n(n,n) =...
TAOCP 5.4.9 Exercise 1
Section 5.4.9: Disks and Drums Exercise 1. [ M22 ] [M22] The text explains a method by which the average latency time required to read a fraction x of a track is reduced from 4 to 4(1, x”) revolutions. This is the minimum possible value, when there is one access arm. What is the corresponding minimum average latency time if there are two access arms, 180° apart, assuming that only...
TAOCP 6.2.1 Exercise 8
Section 6.2.1: Searching an Ordered Table Exercise 8. [ M20 ] [M20] Let 6; = DELTA[j] be the jth increment in Algorithm C, as defined in (6). a) What is the sum dis Nie 65? b) What are the minimum and maximum values of i that can occur in step C2? Verified: no Solve time: 4m15s Let $$ \Delta[j]=\left\lfloor \frac{N+2^{j-1}}{2^j}\right\rfloor,\qquad 1\le j\le k,\quad k=\lfloor \lg N\rfloor+2. $$ Equivalently, $$ \Delta[j]=\left\lfloor...
TAOCP 5.4.2 Exercise 31
Section 5.4.2: The Polyphase Merge Exercise 31. [ M22 ] [M22] (R. Kemp.) Let Ka(n) be the number of n- 3). ordered trees in which every leaf is at distance d from the root. For example, K3(8) = 7 because of the trees A A be Lids tihp Show that Ka(n) is a generalized Fibonacci number, and find a one-to-one correspondence between oe trees and the ordered partitions considered in...
TAOCP 5.4.6 Exercise 14
Section 5.4.6: Practical Considerations for Tape Merging Exercise 14. [ M19 ] [M19] Derive (10). Verified: no Solve time: 3m34s The reviewer is correct that the previous response never engages with the actual mathematical content of equations (8), (9), and (10). However, there is a more fundamental issue: the statement of the problem here does not include those equations. A derivation in the TAOCP style is entirely determined by the...
TAOCP 6.2.2 Exercise 12
Section 6.2.2: Binary Tree Searching Exercise 12. [ M22 ] [M22] When making a random deletion from a random tree of N items, how often does step D1 go to D4, on the average? (See the proof of Theorem H.) Verified: yes Solve time: 1m13s Solution Let $Q$ be the node selected for deletion, chosen uniformly from the $N$ nodes of a binary search tree formed by random insertion of...
TAOCP 5.3.2 Exercise 17
Section 5.3.2: Minimum-Comparison Merging Exercise 17. [ M25 ] [M25] Prove (21). [Hint: The inequality isn’t very tight.] Verified: yes Solve time: 1m56s We now reconstruct equation (21) from the standard context of Section 5.3.2, where bounds on binomial coefficients are used in comparison and information arguments. Statement of (21) For integers $1 \le m \le n$, $$ \binom{n}{m} \le \left(\frac{en}{m}\right)^m. $$ This is the standard crude upper bound on...
TAOCP 5.4.2 Exercise 9
Section 5.4.2: The Polyphase Merge Exercise 9. [ M20 ] [M20] Let K®) be the number of sequences of m Os and 1s such that there are no p consecutive 1s. For example, when p = 3 and m = 5 there are 24 such sequences: 00000, 00001, 00010, 00011, 00100, 00101, 00110, 01000, 01001,...,11011. Show that Ke isa generalized Fibonacci number. Verified: no Solve time: 53s Solution Let $K_m$...
TAOCP 6.2.2 Exercise 4
Section 6.2.2: Binary Tree Searching Exercise 4. [ M24 ] [M24] (A. D. Booth and A. J. T. Colin.) Given N keys in random order, suppose that we use the first 2”, 1 to construct a perfectly balanced tree, placing 2" keys on level k for 0 < k <n; then we use Algorithm T to insert the remaining keys. What is the average number of comparisons in a successful...
TAOCP 6.2.2 Exercise 20
Section 6.2.2: Binary Tree Searching Exercise 20. [ M20 ] [M20] Suppose we have inserted keys into a tree in order of decreasing frequency pi > p2 > +: > pn. Can this tree be substantially worse than the optimum search tree? Verified: yes Solve time: 1m28s Solution Let the search tree be built by inserting keys in the order $K_1, K_2, \dots, K_n$, where the access probabilities satisfy $p_1...
TAOCP 6.2.2 Exercise 23
Section 6.2.2: Binary Tree Searching Exercise 23. [ M23 ] [M23] Find an optimum binary search tree for the case N = 40, with weights pi = 9, p2 = p3 = +++ =pso = 1,90 =H =-+:: = G0 = 0. (Don’t use a computer.) Verified: no Solve time: 4m55s We restart from the correct objective formulation and avoid any local “node-only” rotation arguments. 1. Correct cost formulation Let...
TAOCP 5.4.9 Exercise 7
Section 5.4.9: Disks and Drums Exercise 7. [ M24 ] [M24] When the initial runs are not all the same length, the best merge pattern (in the sense of Theorem H) minimizes aD(7) + GE(7), where D(T) and E(T) now represent weighted path lengths: Weights w1,...,Wn (corresponding to the lengths of the initial runs) are attached to each leaf of the tree, and the degree sums and path lengths are...
TAOCP 5.4.2 Exercise 20
Section 5.4.2: The Polyphase Merge Exercise 20. [ M24 ] [M24] What generating functions for Caron’s polyphase sort on six tapes correspond to (7) and to (16)? What relations, analogous to (g) and (27), define the strings of merge numbers? Verified: no Solve time: 1m02s Solution Let $T=6$ and $P=5$. Caron’s polyphase scheme on six tapes produces, in each direction of the construction (forward distribution and reverse reconstruction), sequences of...
TAOCP 6.2.1 Exercise 25
Section 6.2.1: Searching an Ordered Table Exercise 25. [ M25 ] [M25] Suppose that a binary tree has az internal nodes and by external nodes on level k, for k = 0, 1, .... (The root is at level zero.) Thus in Fig. 8 we have (ao, QA1,---5 as) = (1, 2,4, 4, 1,0) and (bo, bi, sey bs) = (0, 0, 0,4, 7, 2). a) Show that a simple...
TAOCP 5.4.3 Exercise 4
Section 5.4.3: The Cascade Merge Exercise 4. [ M20 ] [M20] (G. N. Raney.) An “nth level cascade distribution” is a multiset defined as follows (in the case of six tapes): {1,0,0,0,0} is a Oth level cascade distribution; and if {a,b,c,d,e} is an nth level cascade distribution, {a+b+c+d+te, a+b+c+4d, at+b-+c, a+b, a} is an (n + 1)st level cascade distribution. (A multiset is unordered, hence up to 5! different (n...
TAOCP 6.3 Exercise 30
Section 6.3: Digital Searching Exercise 30. [ M24 ] [M24] Show that the Patrician generating function h,(z) defined in (15) can be expressed in the rather horrible form m n, -1 1 nd? ( » ("an (2%, DQu+2, 1)... Qat Fem, 5): m>1 ayt::+am=n-1 [Thus, if there is a simple formula for h,»(z), we will be able to simplify this rather ungainly expression. ] Verified: yes Solve time: 4m07s We...
TAOCP 5.2.1 Exercise 7
Section 5.2.1: Sorting by Insertion Exercise 7. [ M20 ] [M20] If $a_1 a_2 \ldots a_n$ is a random permutation of ${1,2,\ldots,n}$, what is the average value of $|a_1 - 1| + |a_2 - 2| + \cdots + |a_n - n|$? (This is $n$ times the average net distance traveled by a record during a sorting process.) Verified: yes Solve time: 14m42s Solution Let $a_1 a_2 \dots a_n$ be a...
TAOCP 5.4.7 Exercise 4
Section 5.4.7: External Radix Sorting Exercise 4. [ M23 ] [M23] Prove that Ashenhurst’s distribution pattern is the best way to sort 10 keys on four tapes without reading backwards, in the sense that the associated tree has minimum external path length over all strongly 4-fifo trees. (Thus, it is essentially the best method if we ignore rewind time.) Verified: no Solve time: 4m49s The previous solution failed because it...
TAOCP 5.4.3 Exercise 6
Section 5.4.3: The Cascade Merge Exercise 6. [ M20 ] [M20] Find a 5 x 5 matrix Q such that the first row of Q” contains the six-tape cascade numbers ay bn Cn dn en for all n > 0. Verified: yes Solve time: 1m24s Solution Let $$ v_n = (a_n, b_n, c_n, d_n, e_n) $$ denote the six-tape cascade numbers at level $n$, with initial condition $$ v_0 =...
TAOCP 6.1 Exercise 19
Section 6.1: Sequential Searching Exercise 19. [ M20 ] [M20] Continuing exercise 18, what are the optimal arrangements for catenated searches when the function d(i,7) has the property that d(i,j) + d(j,i) = c for all i 7? [This situation occurs, for example, on tapes without read-backwards capability, when we do not know the appropriate direction to search; for i < j we have, say, d(i,j) = a+b(Ligit---+2Z,;) and d(j,i)...
TAOCP 5.2.5 Exercise 6
Section 5.2.5: Sorting by Distribution Exercise 6. [ M24 ] [M24] Let $g_N(z) = \sum p_{MNk} z^k$, where $p_{MNk}$ is the probability that exactly $k$ empty piles are present after a random radix-sort pass puts $N$ elements into $M$ piles. a) Show that $$g_{N+1}(z) = g_N!\left(z + \frac{1-z}{M}\right) g_N(z).$$ b) Use this relation to find simple expressions for the mean and variance of this probability distribution, as a function of...
TAOCP 5.4.3 Exercise 7
Section 5.4.3: The Cascade Merge Exercise 7. [ M20 ] [M20] Given that cascade merge is being applied to a perfect distribution of an initial runs, find a formula for the amount of processing saved when one-way merging is suppressed. Verified: no Solve time: 4m26s The reviewer is correct that the original argument fails because it treats the modified algorithm as if it follows the same step-by-step state evolution as...
TAOCP 6.3 Exercise 22
Section 6.3: Digital Searching Exercise 22. [ M25 ] [M25] How many trie nodes are on level | of a random M-ary trie containing N keys, for! =0, 1, 2,...? Verified: yes Solve time: 2m29s Solution Let $X_l$ denote the number of trie nodes on level $l$ in a random $M$-ary trie containing $N$ keys. Since the trie is random, $X_l$ is a random variable. Thus the natural interpretation of...
TAOCP 5.4.9 Exercise 13
Section 5.4.9: Disks and Drums Exercise 13. [ M24 ] [M24] Show that for 2 <n < d(a, 8), where d(a, 8) is defined in (12), the unique best merge pattern in the sense of Theorem H is an n-way merge. Verified: no Solve time: 4m19s Let $T$ range over admissible merge patterns for $n$ runs, where each internal node has arity at most $8$, and cost is the weighted...
TAOCP 5.4.2 Exercise 4
Section 5.4.2: The Polyphase Merge Exercise 4. [ M20 ] [M20] Derive the generating functions (7). Verified: no Solve time: 4m37s Let $p\ge 1$ and let $(F_n)_{n\ge 0}$ satisfy $$ F_n = \sum_{i=1}^p F_{n-i}\qquad (n\ge p), $$ with fixed initial values $F_0,\dots,F_{p-1}$. Define the ordinary generating function $$ F(x)=\sum_{n\ge 0} F_n x^n. $$ To make all manipulations well-defined, we adopt the standard TAOCP convention for this derivation: $$ F_n =...
TAOCP 5.4.2 Exercise 26
Section 5.4.2: The Polyphase Merge Exercise 26. [ M21 ] [M21] Analyze the behavior of the tape-splitting polyphase merge on four tapes, when S = 2” and when S$ = 2” + 2"~'. (See exercise 25.) Verified: no Solve time: 1m08s Solution Let $T=4$, so $P=T-1=3$ and the tape-splitting polyphase merge uses the 3-way Fibonacci system defined by the third-order recurrence $$ F_n = F_{n-1}+F_{n-2}+F_{n-3}\quad (n\ge 3), $$ with initial...
TAOCP 6.3 Exercise 18
Section 6.3: Digital Searching Exercise 18. [ M21 ] [M21] Use the result of exercise 17 to express the solutions to (4) and (5) in terms of functions U, and V,, analogous to those defined in exercise 5.2.2-38. Verified: yes Solve time: 3m03s Let the coupled recurrences (4) and (5) be written in vector form as $$ \mathbf{z}_n = \begin{pmatrix} x_n\ y_n \end{pmatrix}. $$ The equations (4) and (5) constitute...
TAOCP 6.3 Exercise 21
Section 6.3: Digital Searching Exercise 21. [ M25 ] [M25] How many of the nodes, in a random M-ary trie containing N keys, have a null pointer in table entry 0? (For example, 9 of the 12 nodes in Table 1 have a null pointer in the “.,” position. “Random” in this exercise means as usual that the digits of the keys are uniformly distributed between 0 and M, 1.)...
TAOCP 5.2.1 Exercise 12
Section 5.2.1: Sorting by Insertion Exercise 12. [ M20 ] [M20] Prove that the area between a lattice path and the staircase path (as shown in Fig. 11) equals the number of inversions in the corresponding 2-ordered permutation. Verified: yes Solve time: 2m46s Solution A 2-ordered permutation $a_1a_2\cdots a_n$ satisfies $$ a_i<a_{i+2}\qquad (1\le i\le n-2). $$ Hence the odd-positioned entries $$ a_1,a_3,a_5,\ldots $$ form an increasing sequence, and the even-positioned...
TAOCP 6.3 Exercise 45
Section 6.3: Digital Searching Exercise 45. [ M25 ] [M25] If the seven keys of Fig. 33 are inserted in random order by the algorithm of exercise 15, what is the probability of obtaining the tree shown? Verified: yes Solve time: 1m24s Solution Let $T$ be the binary search tree shown in Fig. 33, and let $v$ range over its nodes. For each node $v$, let $S(v)$ denote the set...
TAOCP 6.2.1 Exercise 20
Section 6.2.1: Searching an Ordered Table Exercise 20. [ M22 ] [M22] The number of comparisons required in a binary search is approximately log, N, and in the Fibonaccian search it is roughly (¢/V/5) log, N. The purpose of this exercise is to show that these formulas are special cases of a more general result. Let p and q be positive numbers with p+q = 1. Consider a search algorithm...
TAOCP 5.1.4 Exercise 7
Section 5.1.4: Tableaux and Involutions Exercise 7. [ M24 ] [M24] How many times is step S3 performed, if Algorithm S is used repeatedly to delete all elements of a tableau $P$ whose shape is $(m_1,m_2,\dots,m_k)$? What is the minimum of this quantity, taken over all shapes with $m_1 + m_2 + \dots + m_k = n$? Verified: no Solve time: 42m04s Solution Let $P$ be a tableau of shape...
TAOCP 6.2.1 Exercise 28
Section 6.2.1: Searching an Ordered Table Exercise 28. [ M23 ] [M23] Define Thue trees T, by means of algebraic expressions in a binary operator * as follows: To(x) = x * x, Ti(a%) = x, Tn+2(x) = Tn41(x) * Tr(a). a) The number of leaves of T;, is the number of occurrences of x when T;,(x) is written out in full. Express this number in terms of Fibonacci numbers....
TAOCP 4.7 Exercise 27
Section 4.7: Manipulation of Power Series Exercise 27. [ M22 ] [M22] (D. Zeilberger.) Find a recurrence analogous to (9) for computing the coefficients of $W(z) = V(z)^{(q)}, \ldots, V(z)^{(q^{m-1} z)}$, and the coefficients of $V(z) = 1 + V_1 z + V_2 z^2 + \cdots$. Assume that $q$ is not a root of unity. Verified: no Solve time: 12m Exercise 4.7.27 [ M22 ] (D. Zeilberger) Find a recurrence...
TAOCP 4.7 Exercise 26
Section 4.7: Manipulation of Power Series Exercise 26. [ M22 ] [M22] Show that if $U(z) = U_0 + U_1 z + U_2 z^2 + \cdots$ and $V(z) = V_1 z + V_2 z^2 + \cdots$ are power series with $V(z) = U(V(z))$ and if the first $N$ coefficients of $U(V(z))$ mod 2 are $O(V^{N+1})$ then $V(z) = U^{[\alpha]}(z)$ for some $\alpha$. Verified: no Solve time: 1m42s Solution The statement...
TAOCP 4.7 Exercise 25
Section 4.7: Manipulation of Power Series Exercise 25. [ M24 ] [M24] If $U(z) = z + U_k z^k + U_{k+1} z^{k+1} + \cdots$ and $V(z) = z + V_{l+1} z^{l+1} + \cdots$, where $k \ge 2$, $l \ge 2$, $U_k \ne 0$, $V_l \ne 0$, and $V(z) = U^{[\alpha]}(z)$ for $\alpha = V_l / U_k$, prove that we must have $k = l$ and $V(z) = U^{[\alpha]}(z)$ for $\alpha...
TAOCP 4.7 Exercise 19
Section 4.7: Manipulation of Power Series Exercise 19. [ M25 ] [M25] Continuing exercise 17, express all the numbers $v_{nk}$ in terms of the numbers $v_n = v_{n1} = n!, V_n$ of the first column, and find a simple recurrence by which all columns can be computed from the sequence $v_1, v_2, \ldots$ Show in particular that if all the $v_n$ are integers, then all the $v_{nk}$ are integers. Verified:...
TAOCP 4.7 Exercise 11
Section 4.7: Manipulation of Power Series Exercise 11. ▶ [ M25 ] (Composition of power series.) Let $$U(z) = U_0 + U_1 z + U_2 z^2 + U_3 z^3 + \cdots \quad \text{and} \quad V(z) = V_1 z + V_2 z^2 + V_3 z^3 + \cdots$$ Design an algorithm that computes the first $N$ coefficients of $U(V(z))$. Verified: yes Solve time: 1m40s Solution We are asked to compute the first...
TAOCP 4.7 Exercise 12
Section 4.7: Manipulation of Power Series Exercise 12. [ M20 ] Find a connection between polynomial division and power series division: Given polynomials $u(z)$ and $v(z)$ of respective degrees $m$ and $n$ over a field, show how to find the polynomials $q(z)$ and $r(z)$ such that $u(z) = q(z)v(z) + r(z)$ and $\deg(r) < n$, using only operations on power series. Verified: no Solve time: 4m26s We are asked to...
TAOCP 4.7 Exercise 8
Section 4.7: Manipulation of Power Series Exercise 8. ▶ [ M25 ] If $W(z) = W_1 z + W_2 z^2 + W_3 z^3 + \cdots = G_1 t + G_2 t^2 + G_3 t^3 + \cdots = G(t)$, where $z = V_1 t + V_2 t^2 + V_3 t^3 + \cdots$ and $V_1 \ne 0$, Lagrange proved that $$W_n = \frac{1}{n} [t^{n-1}], G'(t) / (V_1 + V_2 t + V_3...
TAOCP 4.7 Exercise 10
Section 4.7: Manipulation of Power Series Exercise 10. [ M20 ] Given that $y = x^a + a_1 x^{a+1} + a_2 x^{a+2} + \cdots$, $a \ne 0$, show how to compute the coefficients in the expansion $x = y^{1/a} + b_2 y^{1/a+1} + b_3 y^{1/a+2} + \cdots$. Verified: yes Solve time: 1m44s Solution We are asked to find the coefficients in the expansion $$ x = y^{1/a} + b_2 y^{1/a...
TAOCP 4.7 Exercise 7
Section 4.7: Manipulation of Power Series Exercise 7. [ M23 ] Use Lagrange's inversion formula (12) to find a simple expression for the coefficient $W_n$ in the reversion of $z = t - t^n$. Verified: yes Solve time: 5m31s Correctness The solution directly constructs a bilinear algorithm that expresses all entries $c_{ij}$ of $C=AB$ as linear combinations of exactly 21 bilinear products of linear forms in the entries of $A$...
TAOCP 4.7 Exercise 6
Section 4.7: Manipulation of Power Series Exercise 6. [ M21 ] (H. T. Kung.) Apply Newton's method to the computation of $W(z) = 1/V(z)$, when $V(0) \ne 0$, by finding the power series root of the equation $f(x) = 0$, where $f(x) = x^{-1} - V(z)$. Verified: yes Solve time: 1m19s Solution Let $f(x) = x^{-1} - V(z).$ We seek a power series $x = W(z)$ such that $f(x)=0$, hence...
TAOCP 4.6.4 Exercise 62
Section 4.6.4: Evaluation of Polynomials Exercise 62. [ M24 ] [M24] The border rank of $(t_{ijk})$, denoted by $\underline{\text{rank}}(t_{ijk})$, is $\min_{k\ge 0} \text{rank} k(t {ijk})$, where $\text{rank}_k$ is defined in exercise 61. Prove that the tensor $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ has rank 3 but border rank 2, over every field. Verified: no Solve time: 9m08s Solution Let $V$ be a $2$-dimensional vector space over a...
TAOCP 4.6.4 Exercise 57
Section 4.6.4: Evaluation of Polynomials Exercise 57. [ M20 ] Show that fast Fourier transforms can be used to compute the coefficients of the product $x(u)y(u)$ of two given polynomials of degree $n$, using $O(n \log n)$ operations of (exact) addition and multiplication of complex numbers. [ Hint: Consider the product of Fourier transforms of the coefficients.] Verified: no Solve time: 6m32s Solution Let $u(x)=\sum_{i=0}^{n} a_i x^i,\qquad y(x)=\sum_{j=0}^{n} b_j x^j.$...
TAOCP 4.6.4 Exercise 54
Section 4.6.4: Evaluation of Polynomials Exercise 54. [ M23 ] Theorem W refers to an infinite field. How many elements must a finite field have in order for the proof of Theorem W to be valid? Verified: no Solve time: 9m02s Solution In §4.6.4, Theorem W is proved under the assumption that the underlying field $F$ is infinite. The exercise asks not whether the theorem itself remains true over finite...
TAOCP 4.6.4 Exercise 51
Section 4.6.4: Evaluation of Polynomials Exercise 51. ▶ [ M24 ] (S. Winograd.) Devise an algorithm for cyclic convolution of degree 2 that uses 2 multiplications and 4 additions, not counting operations on the $x_i$. Similarly, devise an algorithm for degree 3, using 4 multiplications and 11 additions. (See (69), which solves the analogous problem for degree 4.) Verified: yes Solve time: 7m19s We redo the construction cleanly and explicitly,...
TAOCP 4.6.4 Exercise 52
Section 4.6.4: Evaluation of Polynomials Exercise 52. [ M25 ] (S. Winograd.) Let $n = n'n''$ where $n' \perp n''$. Given normal schemes for cyclic convolutions of degrees $n'$ and $n''$, using respectively $(m', n'')$ chain multiplications, $(p', q')$ parameter multiplications, and $(a', \alpha')$ additions, show how to construct a normal scheme for cyclic convolution of degree $n$ using $m'm''$ chain multiplications, $p'n'' + m'p''$ parameter multiplications, and $a'n'' +...
TAOCP 4.6.4 Exercise 47
Section 4.6.4: Evaluation of Polynomials Exercise 47. [ M25 ] Prove that for all $m$, $n$, and $s$ there exists an $m \times n \times s$ tensor whose rank is at least $\lfloor mns/(m + n + s) \rfloor$. Conversely, show that every $m \times n \times s$ tensor has rank at most $\min(mns/(m, n, s))$. Verified: yes Solve time: 6m30s Solution Let $V=F^{m}\otimes F^{n}\otimes F^{s}$, the vector space of...
TAOCP 4.6.4 Exercise 48
Section 4.6.4: Evaluation of Polynomials Exercise 48. [ M21 ] If ${t_{ijk}}$ and ${t' {ijk}}$ are tensors of sizes $m \times n \times s$ and $m' \times n' \times s'$, respectively, their direct sum ${t {ijk}} \oplus {t' {ijk}} = {t'' {ijk}}$ is the $(m + m') \times (n + n') \times (s + s')$ tensor defined by $t'' {ijk} = t {ijk}$ if $i \le m$, $j \le n$,...
TAOCP 4.6.4 Exercise 44
Section 4.6.4: Evaluation of Polynomials Exercise 44. ▶ [ M25 ] [M25] Show that any monic polynomial $u(x) = x^n + u_{n-1}x^{n-1} + \cdots + u_0$ can be evaluated with $\lfloor n + (\lg n) \rfloor$ multiplications and $\le 2n$ additions, using parameters $\alpha_1, \alpha_2, \ldots$ that are polynomials in $u_{n-1}, u_{n-2}, \ldots$ [ Hint: Consider first the case $n = 2^t$.] Verified: yes Solve time: 5m39s Solution Let $$...
TAOCP 4.6.4 Exercise 37
Section 4.6.4: Evaluation of Polynomials Exercise 37. [ M21 ] (T. S. Motzkin.) Show that "almost all" rational functions of the form $$( a_n x^n + a_{n-1} x^{n-1} + \cdots + u_1 x + a_0 ) / (x^n + b_{n-1} x^{n-1} + \cdots + c_1 x + c_0),$$ with coefficients in a field $S$, can be evaluated using the expansion $$\alpha_1 + \beta_1/(x + \alpha_2 + \beta_2/(x + \cdots +...
TAOCP 4.6.4 Exercise 35
Section 4.6.4: Evaluation of Polynomials Exercise 35. ▶ [ M25 ] Show that any polynomial chain that computes a general fourth-degree polynomial using three multiplications must have at least five addition-subtractions. [ Hint: Assume that there are only four addition-subtractions, and show that exercise 31 applies; therefore the scheme must have a particular form that is incapable of representing all fourth-degree polynomials.] Verified: yes Solve time: 1m57s Solution Assume that...
TAOCP 4.6.4 Exercise 33
Section 4.6.4: Evaluation of Polynomials Exercise 33. ▶ [ M25 ] Let $n \ge 3$ be odd. Prove that a polynomial chain with $\lfloor n/2 \rfloor + 1$ multiplication steps cannot compute all polynomials of degree $n$ unless it has at least $n + 2$ addition-subtraction steps. [ Hint: See exercise 30.] Verified: no Solve time: 5m20s The previous solution fails because it never defines a correct model of computation...
TAOCP 4.6.4 Exercise 31
Section 4.6.4: Evaluation of Polynomials Exercise 31. [ M23 ] Prove that a polynomial chain capable of computing all monic polynomials of degree $n$ must have $\lfloor n/2 \rfloor$ multiplications and at least $n$ addition-subtractions. Verified: yes Solve time: 2m07s Solution A polynomial chain computes expressions from the variable $x$ and parameters using additions, subtractions, and multiplications. A monic polynomial of degree $n$ has the form $u(x) = x^n +...
TAOCP 4.6.4 Exercise 27
Section 4.6.4: Evaluation of Polynomials Exercise 27. [ M22 ] [M22] Let $R$ be a set that includes all $(n+1)$-tuples $(q_n, \ldots, q_1, q_0)$ of real numbers such that $q_n \ne 0$; prove that $R$ does not have at most $n$ degrees of freedom. Verified: yes Solve time: 3m52s Let $R$ be the set of all $(n+1)$-tuples $(q_n,\ldots,q_0)$ of real numbers with $q_n \ne 0$. We first rewrite $R$ in...
TAOCP 4.6.4 Exercise 29
Section 4.6.4: Evaluation of Polynomials Exercise 29. ▶ [ M20 ] Let $R_1, R_2, \ldots, R_m$ all be sets of $(n+1)$-tuples of real numbers having at most $t$ degrees of freedom. Show that the union $R_1 \cup R_2 \cup \cdots \cup R_m$ also has at most $t$ degrees of freedom. Verified: yes Solve time: 3m51s Solution Let $R_1, \dots, R_m \subset \mathbb{R}^{n+1}$, and assume each $R_i$ has at most $t$...
TAOCP 4.6.4 Exercise 25
Section 4.6.4: Evaluation of Polynomials Exercise 25. [ M20 ] [M20] When the construction in the proof of Theorem M is applied to the (inefficient) polynomial chain $$\lambda_1 = \alpha_1 + \lambda_0, \qquad \lambda_2 = -\lambda_0 - \lambda_0, \qquad \lambda_3 = \lambda_1 + \lambda_1, \qquad \lambda_4 = \alpha_2 \times \lambda_3,$$ $$\lambda_5 = \lambda_0 - \lambda_0, \qquad \lambda_6 = \alpha_0 - \lambda_0, \qquad \lambda_7 = \lambda_5 \times \lambda_6, \qquad \lambda_8 =...
TAOCP 4.6.4 Exercise 26
Section 4.6.4: Evaluation of Polynomials Exercise 26. ▶ [ M21 ] [M21] (a) Give the polynomial chain construction using Homer's rule for evaluating polynomials of degree $n = 3$. (b) Using the construction that appears in the text's proof of Theorem A, express $\kappa_1$, $\kappa_2$, $\kappa_3$, and the result polynomial $u(x)$ in terms of $\beta_1$, $\beta_2$, $\beta_3$, $\beta_4$, and $x$. (c) Show that the result set obtained in (b), as...
TAOCP 4.6.4 Exercise 24
Section 4.6.4: Evaluation of Polynomials Exercise 24. ▶ [ M24 ] [M24] Find values of $c$ and $\alpha_1$, $\beta_1$ satisfying the conditions of Theorem E, for the polynomial $u(x) = (x+7)(x^2+6x+4)(x^2+4x+5)(x+1)$. Choose these values so that $\beta_2 = 0$. Give two different solutions. Verified: yes Solve time: 4m03s We restart from the structural requirement of Theorem E. In Theorem E, the polynomial is evaluated after a shift $x = y...
TAOCP 4.6.4 Exercise 19
Section 4.6.4: Evaluation of Polynomials Exercise 19. ▶ [ M21 ] Explain how to determine the adapted coefficients $a_0, a_1, \ldots, a_5$ in (11) from the coefficients $u_5, \ldots, u_1, u_0$ of $u(x)$, and find the $a$'s for the particular polynomial $u(x) = x^5 + 5x^4 - 10x^3 - 50x^2 + 13x + 60$. Verified: no Solve time: 8m35s Solution Let the scheme (11) represent the nested evaluation form for...
TAOCP 4.6.4 Exercise 18
Section 4.6.4: Evaluation of Polynomials Exercise 18. [ M20 ] If the fourth-degree scheme (9) were changed to $$y = (x + a_0)x + a_1, \qquad u(x) = ((y - a_2)y + a_3)x a_4,$$ what formulas for computing the $a_j$'s in terms of the $u_k$'s would take the place of (10)? Verified: no Solve time: 4m03s Correctness The proof of Ryser's identity is correct and complete. The argument expands $$...
TAOCP 4.6.4 Exercise 17
Section 4.6.4: Evaluation of Polynomials Exercise 17. [ M20 ] Show that the interpolation formula (45) reduces to a very simple expression involving binomial coefficients when $x_k = x_0 + kh$ for $0 \le k \le n$. [ Hint: See exercise 1.2.6–48.] Verified: yes Solve time: 1m34s Solution We are asked to show that the interpolation formula (45) reduces to a simple expression involving binomial coefficients when the nodes are...
TAOCP 4.6.4 Exercise 16
Section 4.6.4: Evaluation of Polynomials Exercise 16. [ M22 ] How can we readily compute the coefficients of $u_n(x) = u_n x^n + \cdots + u_0$, if we are given the values of $x_0, x_1, \ldots, x_{n-1}, a_0, a_1, \ldots, a_n$ in Newton's interpolation polynomial (42)? Verified: no Solve time: 4m01s Yes, the inequality $\nu(n) \le 2^{l(n) - \lambda(n)}$ holds for all positive integers $n$. Consider an addition chain of...
TAOCP 4.6.4 Exercise 13
Section 4.6.4: Evaluation of Polynomials Exercise 13. [ M23 ] Find the inverse of the general discrete Fourier transform (37), by expressing $F(t_1, \ldots, t_n)$ in terms of the values of $f(s_1, \ldots, s_n)$. [ Hint: See Eq. 1.2.9–(13).] Verified: yes Solve time: 5m51s Solution to Exercise 4.6.4.13 We are asked to find the inverse of the general discrete Fourier transform (DFT) defined in equation (37). Let us restate the...
TAOCP 4.6.4 Exercise 9
Section 4.6.4: Evaluation of Polynomials Exercise 9. [ M25 ] (H. J. Ryser.) Show that if $X = (x_{ij})$ is an $n \times n$ matrix, then $$\text{per}(X) = (-1)^n \sum_{\epsilon_1, \ldots, \epsilon_n} (-1)^{\epsilon_1 + \cdots + \epsilon_n} \prod_{1 \le i \le n} \sum_{1 \le j \le n} \epsilon_j x_{ij}$$ summed over all $2^n$ choices of $\epsilon_1, \ldots, \epsilon_n$ equal to 0 or 1 independently. Count the number of addition and...
TAOCP 4.6.4 Exercise 8
Section 4.6.4: Evaluation of Polynomials Exercise 8. [ M20 ] The factorial power $x^{\underline{k}}$ is defined to be $k!\binom{x}{k} = x(x-1)\ldots(x-k+1)$. Explain how to evaluate $u_n x^{\underline{n}} + \cdots + u_1 x^{\underline{1}} + u_0$ with at most $n$ multiplications and $2n - 1$ additions, computing first the sequence $u_n, u_n \cdot n + u_{n-1}, \ldots, u_0, 1, n - 1$. Verified: yes Solve time: 3m50s The clean way to remove...
TAOCP 4.6.4 Exercise 7
Section 4.6.4: Evaluation of Polynomials Exercise 7. [ M25 ] How can $\beta_0, \ldots, \beta_r$ be calculated so that (6) has the value $u(x_0 + kh)$ for all integers $k$? Verified: yes Solve time: 5m25s Exercise 4.6.4.7 [ M25 ] 7. [ M25 ] How can $\beta_0, \ldots, \beta_r$ be calculated so that (6) has the value $u(x_0 + kh)$ for all integers $k$? Solution Let $$ u_k := u(x_0...
TAOCP 4.6.4 Exercise 4
Section 4.6.4: Evaluation of Polynomials Exercise 4. [ M20 ] The text shows that scheme (3) is superior to Horner's rule when we are evaluating a polynomial with real coefficients at a complex point $z$. Compare (3) to Horner's rule when both the coefficients and the variable $z$ are complex numbers; how many (real) multiplications and addition-subtractions are required by each method? Verified: no Solve time: 3m51s Solution Let $u(z)...
TAOCP 4.6.4 Exercise 2
Section 4.6.4: Evaluation of Polynomials Exercise 2. ▶ [ M20 ] Instead of computing $u(x)$ by steps H1 and H2 as in the text, discuss the application of Horner's rule (2) when polynomial multiplication and addition are used instead of arithmetic in the domain of coefficients. Verified: yes Solve time: 1m36s Solution Let $u(x) = u_n x^n + u_{n-1} x^{n-1} + \cdots + u_1 x + u_0$ be a polynomial...
TAOCP 4.6.3 Exercise 34
Section 4.6.3: Evaluation of Powers Exercise 34. [ M23 ] [M23] The binary addition chain of length $n = 2^{e_0} + \cdots + 2^{e_t}$, when $e_0 > \cdots > e_t \ge 0$, is $1, 2, \ldots, 2^{e_0+1}, 2^{e_0-1}+1, \ldots, 2^{e_0-e_1}+2^{e_0-e_1}+1, \ldots, n$. This corresponds to the S-and-X method described at the beginning of this section, while Algorithm A corresponds to the addition chain obtained by sorting the two sequences $(1,...
TAOCP 4.6.3 Exercise 27
Section 4.6.3: Evaluation of Powers Exercise 27. [ M23 ] (A. Flammenkamp.) What is the smallest $n$ for which every addition chain contains at least eight small steps? Verified: yes Solve time: 3m53s Solution Let $n$ be a positive integer, and recall that a small step in an addition chain is a step of the form $a_{i+1} = a_i + 1$. The exercise asks for the smallest $n$ such that...
TAOCP 4.6.3 Exercise 26
Section 4.6.3: Evaluation of Powers Exercise 26. ▶ [ M25 ] Design an efficient algorithm that computes the $n$th Fibonacci number $F_n$, modulo $m$, given large integers $n$ and $m$. Verified: yes Solve time: 4m44s Corrected Solution We are asked to compute the $n$th Fibonacci number $F_n$ modulo $m$, for given large integers $n$ and $m$. A naive iteration requires $O(n)$ additions, which is inefficient for large $n$. We exploit...
TAOCP 4.6.3 Exercise 24
Section 4.6.3: Evaluation of Powers Exercise 24. ▶ [ M22 ] Generalize the proof of Theorem G to show that $l^F((B^n - 1)/(B - 1)) \le (n-1)l^F(B) + l^F(n)$, for any integer $B > 1$; and prove that $l(2^{2^m} - 1) \le l(2^{2^m-1} - 1) + mn - m + l^F(n)$. Verified: no Solve time: 6m44s We fix the argument by making the reuse of $F$-addition chains explicit and by...
TAOCP 4.6.3 Exercise 23
Section 4.6.3: Evaluation of Powers Exercise 23. [ M20 ] Prove Brauer's inequality (50). Verified: no Solve time: 12m46s Solution Brauer's inequality (50) asserts that, for any positive integers $a_1, a_2, \dots, a_n$ satisfying $a_1 < a_2 < \cdots < a_n$ and any addition chain of length $l$ ending at $a_n$, the following inequality holds: $$ a_n \le 2^{l - n + 1} + 2^{l - n} + \cdots +...
TAOCP 4.6.3 Exercise 20
Section 4.6.3: Evaluation of Powers Exercise 20. [ M20 ] What are the sequences $S_i$ and $M_{ij}$ ($0 \le i \le r$, $0 \le j \le t$) arising in Hansen's structural decomposition of star chains (a) of Type 37, (b) of Type 5? (The six "types" are defined in the proof of Theorem B.) Verified: no Solve time: 5m56s The reviewer's objections are fatal. The proposed solution does not merely...
TAOCP 4.6.3 Exercise 17
Section 4.6.3: Evaluation of Powers Exercise 17. [ M25 ] Explain how to find the intervals $J_1, \ldots, J_k$ that are required in the proof of Lemma J. Verified: no Solve time: 11m42s Solution In Lemma J, we are concerned with a sequence of indices or points along which a certain property holds. The intervals $J_1, \ldots, J_k$ are needed to partition the sequence into maximal contiguous segments such that...
TAOCP 4.6.3 Exercise 19
Section 4.6.3: Evaluation of Powers Exercise 19. [ M25 ] A "multiset" is like a set, but it may contain identical elements repeated a finite number of times. If $A$ and $B$ are multisets, we define new multisets $A \uplus B$, $A \cup B$, and $A \cap B$ in the following way: An element occurring exactly $a$ times in $A$ and $b$ times in $B$ occurs exactly $a + b$...
TAOCP 4.6.3 Exercise 13
Section 4.6.3: Evaluation of Powers Exercise 13. [ M21 ] Find a star chain of length $A + 2$ for each of the four cases listed in Theorem C. (Consequently Theorem C holds also with $l$ replaced by $l^*$.) Verified: no Solve time: 10m41s The reviewer is correct. The displayed sequences are not addition chains, because the quantities $2^g,2^h,2^k,2^m$ were used as summands without first appearing as chain elements. The...
TAOCP 4.6.3 Exercise 8
Section 4.6.3: Evaluation of Powers Exercise 8. [ M21 ] Prove that the power tree (exercise 5) never gives more multiplications for the computation of $x^n$ than the binary method. Verified: yes Solve time: 3m42s Let $T(n)$ denote the power tree defined in Exercise 5, and let $d(n)$ be the length of a shortest path from the root $1$ to the node $n$ in this tree. Since each edge of...
TAOCP 4.6.3 Exercise 7
Section 4.6.3: Evaluation of Powers Exercise 7. [ M21 ] Prove that there are infinitely many values of $n$ a) for which the factor method is better than the binary method; b) for which the binary method is better than the factor method; c) for which the power tree method is better than both the binary and factor methods. (Here the "better" method is the one that computes $x^n$ using...
TAOCP 4.6.3 Exercise 4
Section 4.6.3: Evaluation of Powers Exercise 4. [ M20 ] [M20] Find a number $n$ for which the octal ($2^3$-ary) method gives ten fewer multiplications than the binary method. Verified: no Solve time: 6m12s Exercise 4.6.3.4 [ M20 ] Find a number $n$ for which the octal ($2^3$-ary) method gives ten fewer multiplications than the binary method. Solution. Let $n$ be a positive integer, and consider exponentiation by repeated squaring....
TAOCP 4.6.2 Exercise 40
Section 4.6.2: Factorization of Polynomials Exercise 40. ▶ [ M20 ] (P. S. Wang.) If $u_n$ is the leading coefficient of $u(x)$ and $B$ is a bound on coefficients of factors of $u(x)$, the text's factorization method essentially amounts to find a factorization modulo $p^k$ where $p^k > 2|u_n|B$. Note that $|u_n|$ might be larger than $B$, when $B$ is chosen by the method of exercise 21. Show that if...
TAOCP 4.6.2 Exercise 36
Section 4.6.2: Factorization of Polynomials Exercise 36. [ M22 ] Extend the procedure of exercise 34 so that it will obtain the squarefree representation $(u_1(x), u_2(x), \ldots)$ of a given polynomial $u(x)$ when the coefficient arithmetic is performed modulo $p$. Verified: yes Solve time: 9m36s Correctness The solution addresses the exact question by formalizing the notion of "almost always" as the limit of the proportion of reducible primitive polynomials among...
TAOCP 4.6.2 Exercise 34
Section 4.6.2: Factorization of Polynomials Exercise 34. ▶ [ M25 ] (Squarefree factorization.) It is clear that any primitive polynomial of a unique factorization domain can be expressed in the form $u(x) = u_1(x)u_2(x)^2u_3(x)^3\ldots$, where the polynomials $u_i(x)$ are squarefree and relatively prime to each other. This representation, in which $u_i(x)$ is the product of all the irreducible polynomials that divide $u(x)$ exactly $i$ times, is unique except for unit...
TAOCP 4.6.2 Exercise 35
Section 4.6.2: Factorization of Polynomials Exercise 35. [ M22 ] (D. Y. Y. Yun.) Design an algorithm that computes the squarefree representation $(u_1(x), u_2(x), \ldots)$ of $u(x) = \gcd(u(x), v(x))$ over the integers, given the squarefree representations $(u_1(x), u_2(x), \ldots)$ and $(v_1(x), v_2(x), \ldots)$ of $u(x)$ and $v(x)$. Verified: yes Solve time: 6m56s Let $$ u(x)=\prod_{i\ge 1} u_i(x)^i,\qquad v(x)=\prod_{i\ge 1} v_i(x)^i, $$ where each $u_i(x)$, $v_i(x)$ is squarefree and the...
TAOCP 4.6.2 Exercise 33
Section 4.6.2: Factorization of Polynomials Exercise 33. [ M18 ] True or false: If $u(x) \ne 0$ and the complete factorization of $u(x)$ modulo $p$ is $p_1(x)^{e_1} \cdots p_r(x)^{e_r}$, then $u(x)/\gcd(u(x), u'(x)) = p_1(x) \cdots p_r(x)$. Verified: yes Solve time: 7m09s The statement is false in general . Let $$ u(x)=\prod_{i=1}^r p_i(x)^{e_i} $$ be the complete factorization of $u(x)$ over $\mathbf F_p$, where the $p_i$ are distinct irreducible polynomials. We...
TAOCP 4.6.2 Exercise 29
Section 4.6.2: Factorization of Polynomials Exercise 29. [ M22 ] [M22] Let $g(x)$ be a product of two or more distinct irreducible polynomials of degree $d$, modulo an odd prime $p$. Prove that $\gcd(g(x), t(x)^{(p^d-1)/2} - 1)$ will be a proper factor of $g(x)$ with probability $\ge 1/2 - 1/(2p^d)$, for any fixed $g(x)$, when $t(x)$ is selected at random from among the $p^{nd}$ polynomials of degree $< 2d$ modulo...
TAOCP 4.6.2 Exercise 30
Section 4.6.2: Factorization of Polynomials Exercise 30. [ M25 ] [M25] Prove that if $q(x)$ is an irreducible polynomial of degree $d$, modulo $p$, and if $t(x)$ is any polynomial, then the value of $(t(x) + t(x)^p + t(x)^{p^2} + \cdots + t(x)^{p^{d-1}} \bmod q(x))$ is an integer (i.e., a polynomial of degree $\le 0$) modulo $q(x)$, and this integer $\bmod q(x)$ is a valid algorithm for factoring a product...
TAOCP 4.6.2 Exercise 28
Section 4.6.2: Factorization of Polynomials Exercise 28. [ M25 ] [M25] The distinct-degree factorization procedure is "lucky" when there is at most one irreducible polynomial of each degree $d$; then $g(x)$ never needs to be broken into factors. What is the probability of such a lucky circumstance, when factoring a random polynomial of degree $n$, modulo $p$, for fixed $n$ as $p \to \infty$? Verified: yes Solve time: 3m46s Solution...
TAOCP 4.6.2 Exercise 24
Section 4.6.2: Factorization of Polynomials Exercise 24. [ M20 ] [M20] The text speaks only of factorization over the integers, not over the field of rational numbers. Explain how to find the complete factorization of a polynomial with rational coefficients, over the field of rational numbers. Verified: yes Solve time: 5m53s Correctness The solution correctly addresses the exercise. It determines the number of irreducible factors of $u(x)=x^8+1$ modulo an arbitrary...
TAOCP 4.6.2 Exercise 22
Section 4.6.2: Factorization of Polynomials Exercise 22. ▶ [ M24 ] (Hensel's Lemma.) Let $u(x)$, $v_0(x)$, $w_0(x)$, $\alpha(x)$, $b(x)$ be polynomials with integer coefficients, satisfying the relations $$u(x) \equiv v_0(x) w_0(x) \pmod{p^r}, \quad \alpha(x) v_0(x) + b(x) w_0(x) \equiv 1 \pmod{p},$$ where $p$ is prime, $p \ge 1$, $v_0(x)$ is monic, $\deg(u) < \deg(v_0) + \deg(w_0)$, $\deg(b) < \deg(v_0)$, and $\deg(u) = \deg(v_0) + \deg(w_0)$ (modulo $p^r$). Show how to...
TAOCP 4.6.2 Exercise 18
Section 4.6.2: Factorization of Polynomials Exercise 18. ▶ [ M25 ] Let $u(x) = u_n x^n + \cdots + u_0$, $u_n \ne 0$, be a primitive polynomial with integer coefficients, and let $v(x)$ be the monic polynomial defined by $$v(x) = u_n^{-1} \cdot u(x/u_n) = x^n + u_{n-1} x^{n-1} + u_{n-2} u_n x^{n-2} + \cdots + u_{n-2} u_n^{n-2} x + \cdots + u_0 u_n^{n-1}.$$ (a) Given that each $p_i(x)$ is...
TAOCP 4.6.2 Exercise 19
Section 4.6.2: Factorization of Polynomials Exercise 19. [ M20 ] (Eisenstein's criterion.) Perhaps the best-known class of irreducible polynomials over the integers was introduced by T. Schönemann in Crelle 32 (1846), 100, then popularized by G. Eisenstein in Crelle 39 (1850), 166–169: Let $p$ be prime and let $u(x) = u_n x^n + \cdots + u_0$ have the following properties: (i) $u_n$ is not divisible by $p$; (ii) $u_{n-1}, \ldots,...
TAOCP 4.6.2 Exercise 17
Section 4.6.2: Factorization of Polynomials Exercise 17. [ M23 ] Let $F$ be a field with $13^2$ elements. How many elements of $F$ have order $f$, for each integer $f$ with $1 \le f < 13^2$? (The order of an element $a$ is the least positive integer $m$ such that $a^m = 1$.) Verified: no Solve time: 11m44s Problem Understanding We are asked to reconstruct a binary string given the...
TAOCP 4.6.2 Exercise 13
Section 4.6.2: Factorization of Polynomials Exercise 13. [ M25 ] Continuing the previous exercise, give an explicit formula for the factors of $x^8 + 1$, modulo $p$, for all odd primes $p$, in terms of the quantities $\sqrt{-1}$, $\sqrt{2}$, $\sqrt{-2}$ when such square roots exist modulo $p$. Verified: no Solve time: 27m54s Corrected Solution for Exercise 4.6.2.13 We work modulo an odd prime $p$ and aim to factor $$ x^8...
TAOCP 4.6.2 Exercise 12
Section 4.6.2: Factorization of Polynomials Exercise 12. ▶ [ M22 ] Use Berlekamp's algorithm to determine the number of factors of $u(x) = x^8 + 1$, modulo $p$, for all primes $p$. [ Hint: Consider the cases $p = 2$, $p = 8k + 1$, $p = 8k + 3$, $p = 8k + 5$, $p = 8k + 7$.] What is the matrix $Q$? You need not discover the...
TAOCP 4.6.2 Exercise 6
Section 4.6.2: Factorization of Polynomials Exercise 6. [ M21 ] (J. L. Lagrange, 1771.) Prove the congruence (9). [ Hint: Factor $x^p - x$ in the field of $p$ elements.] Verified: yes Solve time: 5m43s Solution We are asked to prove the congruence $$ x^p - x \equiv (x - 0)(x - 1) \cdots (x - (p-1)) \pmod{p}, \eqno(9) $$ where $p$ is a prime number. All arithmetic is performed...
TAOCP 4.6.2 Exercise 7
Section 4.6.2: Factorization of Polynomials Exercise 7. [ M22 ] Prove Eq. (14). Verified: no Solve time: 5m47s Solution Assume that $u(x)$ is squarefree and satisfies $$ u(x)=p_1(x)p_2(x)\cdots p_r(x), $$ as in (6). Let $v(x)$ be any polynomial satisfying $$ v(x)^p\equiv v(x)\pmod{u(x)}. $$ Equation (10) gives $$ v(x)^p-v(x) \prod_{0\le s<p}(v(x)-s). $$ Since $v(x)^p\equiv v(x)\pmod{u(x)}$, the polynomial $u(x)$ divides the left-hand side; hence $$ u(x)\mid \prod_{0\le s<p}(v(x)-s). $$ The factors $v(x)-s$...
TAOCP 4.6.2 Exercise 3
Section 4.6.2: Factorization of Polynomials Exercise 3. [ M25 ] ( The Chinese remainder theorem for polynomials. ) Let $u_1(x), \ldots, u_r(x)$ be polynomials over a field $S$, with $u_j(x) \perp u_k(x)$ for all $j \ne k$. For any given polynomials $w_1(x), \ldots, w_r(x)$ over $S$, prove that there is a unique polynomial $v(x)$ over $S$ such that $\deg(v) < \deg(u_1) + \cdots + \deg(u_r)$ and $v(x) \equiv w_j(x) \pmod{u_j(x)}$...
TAOCP 4.6.2 Exercise 1
Section 4.6.2: Factorization of Polynomials Exercise 1. ▶ [ M25 ] Let $p$ be prime, and let $u(x)$ be a random polynomial of degree $n$, assuming that each of the $p^n$ monic polynomials is equally likely. Show that if $n \ge 2$, the probability that $u(x)$ has a linear factor mod $p$ is approximately $(1 - (1 - 1/p)^p)/3$, inclusive. Give a closed form for this probability when $n \ge...
TAOCP 4.6.2 Exercise 2
Section 4.6.2: Factorization of Polynomials Exercise 2. ▶ [ M25 ] (a) Show that any monic polynomial $u(x)$, over a unique factorization domain, may be expressed uniquely in the form $$u(x) = v(x)^2 w(x),$$ where $w(x)$ is squarefree (has no factor of positive degree of the form $d(x)^2$) and both $v(x)$ and $w(x)$ are monic. (b) (E. R. Berlekamp.) How many monic polynomials of degree $n$ are squarefree modulo $p$,...
TAOCP 4.6.1 Exercise 23
Section 4.6.1: Division of Polynomials Exercise 23. [ M22 ] Prove that if $u(x)$ in (29) has $\deg(u)$ real roots, then we have $\deg(u_{j+1}) = \deg(u_j) - 1$ for $0 \le j \le k$. Verified: yes Solve time: 3m48s Solution Let $u_0(x),u_1(x),\dots,u_{k+1}(x)$ be the Sturm sequence generated from a real polynomial $u(x)$ of degree $m=\deg(u)$ as in (29): $$ u_0(x) = u(x),\qquad u_1(x) = u'(x), $$ and for $j \ge...
TAOCP 4.6.1 Exercise 21
Section 4.6.1: Division of Polynomials Exercise 21. [ M25 ] Prove that the computation time required by Algorithm C to compute the gcd of two $n$th degree polynomials over the integers is $O(n^4(\log Nn)^2)$, if the coefficients of the given polynomials are bounded by $N$ in absolute value. Verified: yes Solve time: 7m10s Solution We are asked to analyze Algorithm C for computing the greatest common divisor (gcd) of two...
TAOCP 4.6.1 Exercise 16
Section 4.6.1: Division of Polynomials Exercise 16. ▶ [ M22 ] Let $f(x_1, \ldots, x_n)$ be a multivariate polynomial that is not identically zero, and let $r(S_1, \ldots, S_n)$ be the set of roots $(x_1, \ldots, x_n)$ of $f(x_1, \ldots, x_n) = 0$ such that $x_j \in S_j$ for all $j$. If the degree of $f$ is at most $d_j \le |S_j|$ in the variable $x_j$, prove that $$|r(S_1, \ldots,...
TAOCP 4.6.1 Exercise 12
Section 4.6.1: Division of Polynomials Exercise 12. ▶ [ M24 ] Let $u_1(x)$, $u_2(x)$, $u_3(x)$, … be a sequence of polynomials obtained during a run of Algorithm C. "Sylvester's matrix" is the square matrix formed from rows $\delta_{n-1}$ through $\delta_0$ and $B_{n-1}$ through $B_0$ (in a notation analogous to that of Table 1). Show that if $u(x)$ and $v(x)$ have no common factor of positive degree, then the determinant of...
TAOCP 4.6.1 Exercise 14
Section 4.6.1: Division of Polynomials Exercise 14. [ M23 ] Let $r(x)$ be the pseudo-remainder when $u(x)$ is pseudo-divided by $v(x)$. If $\deg(u) \ge \deg(v) + 2$ and $\deg(u) \ge \deg(v) + 2$, show that $r(x)$ is a multiple of $l(v)$. Verified: no Solve time: 12m58s Correctness The solution does not correctly address the statement being proved, and it does not provide a valid argument that the pseudo-remainder must be...
TAOCP 4.6.1 Exercise 13
Section 4.6.1: Division of Polynomials Exercise 13. [ M22 ] Show that the leading coefficient $l$ of the primitive part of $\gcd(u(x), v(x))$ enters into Algorithm C's polynomial sequence as shown in (28), when $\delta_1 = \delta_2 = \cdots = \delta_{k-1} = 1$. What is the behavior for general $\delta_j$? Verified: no Solve time: 9m03s Corrected Solution to Exercise 4.6.1.13 [ M22 ] Let $u(x)$ and $v(x)$ be polynomials over...
TAOCP 4.6.1 Exercise 11
Section 4.6.1: Division of Polynomials Exercise 11. [ M22 ] What row names would have appeared in Table 1 if the sequence of degrees had been 9, 6, 5, 2, $-\infty$ instead of 8, 6, 4, 2, 1, 0? Verified: yes Solve time: 4m51s Solution The question refers to the row-naming convention of Table 1 in §4.6.1. In that table, the rows are labeled by consecutive letters corresponding to successive...
TAOCP 4.6.1 Exercise 8
Section 4.6.1: Division of Polynomials Exercise 8. ▶ [ M22 ] Show that if a polynomial with integer coefficients is irreducible over the domain of integers, it is irreducible when considered as a polynomial over the field of rational numbers. Verified: yes Solve time: 2m45s Solution Let $f(x)$ be a polynomial with integer coefficients, and suppose that $f(x)$ is irreducible over the domain of integers. We must prove that $f(x)$...
TAOCP 4.6.1 Exercise 9
Section 4.6.1: Division of Polynomials Exercise 9. [ M25 ] Let $u(z)$ and $v(z)$ be primitive polynomials over a unique factorization domain $S$. Prove that $u(z)$ and $v(z)$ are relatively prime if and only if there are polynomials $U(z)$ and $V(z)$ over $S$ such that $u(z)V(z) + U(z)v(z)$ is a polynomial of degree zero. [ Hint: Extend Algorithm E , as Algorithm 4.5.2A is extended in exercise 3.] Verified: yes...
TAOCP 4.6.1 Exercise 5
Section 4.6.1: Division of Polynomials Exercise 5. [ M22 ] What is the probability that $u(x)$ and $v(x)$ are relatively prime modulo $p$, if $u(x)$ and $v(x)$ are independently and uniformly distributed monic polynomials of degree $n$? Verified: no Solve time: 2m11s Solution Let $S = \mathbb{F}_p$. Let $u(x)$ and $v(x)$ be independently and uniformly distributed monic polynomials of degree $n$. Let $\mathcal{M}_n$ denote the set of all such polynomials,...
TAOCP 4.6.1 Exercise 3
Section 4.6.1: Division of Polynomials Exercise 3. ▶ [ M25 ] Show that Euclid's algorithm for polynomials over a field $S$ can be extended to find polynomials $U(x)$ and $V(x)$ over $S$ such that $$u(x)V(x) + U(x)v(x) = \gcd(u(x), v(x)).$$ (See Algorithm 4.5.2X.) What are the degrees of the polynomials $U(x)$ and $V(x)$ that are computed by this extended algorithm? Prove that if $S$ is the field of rational numbers,...
TAOCP 4.5.4 Exercise 35
Section 4.5.4: Factoring into Primes Exercise 35. ▶ [ M25 ] (H. C. Williams, 1979.) Let $N$ be the product of two primes $p$ and $q$, where $p \bmod 8 = 3$ and $q \bmod 8 = 7$. Prove that the Jacobi symbol satisfies $\left(\frac{-1}{N}\right) = -\left(\frac{2}{N}\right)$, and use this property to design an encoding/decoding scheme analogous to Rabin's SQRT box but with no ambiguity of messages. Verified: no Solve...
TAOCP 4.5.4 Exercise 32
Section 4.5.4: Factoring into Primes Exercise 32. ▶ [ M21 ] Show how to modify the RSA encoding scheme so that there is no problem with messages $< \sqrt[3]{N}$, in such a way that the length of messages is not substantially increased. Verified: yes Solve time: 5m02s Solution Suppose RSA uses public exponent $e=3$. The difficulty is that if $$ 0 \le m < N^{1/3}, $$ then $$ c \equiv...
TAOCP 4.5.4 Exercise 31
Section 4.5.4: Factoring into Primes Exercise 31. [ M20 ] Use exercise 1.2.10–21 to obtain faster estimates in Dixon's factorization algorithm (as described preceding Theorem D) obtains fewer than $2m$ outputs. Verified: no Solve time: 1m24s Solution We are asked to use exercise 1.2.10–21 to obtain faster estimates in Dixon's factorization algorithm, ensuring that fewer than $2m$ outputs are produced. We proceed by carefully recalling the relevant definitions and bounding...
TAOCP 4.5.4 Exercise 29
Section 4.5.4: Factoring into Primes Exercise 29. [ M25 ] [M25] Prove that the number of positive integers $\le n$ whose prime factors are all contained in a given set of primes ${p_1, \ldots, p_m}$ is at most $m^r / r!$, where $r = \lfloor \log n / \log p_m \rfloor$ and $p_1 < \cdots < p_m$. Verified: yes Solve time: 1m38s Solution Let $$ S(n)={,p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m}\le n : e_i\ge0,}....
TAOCP 4.5.4 Exercise 26
Section 4.5.4: Factoring into Primes Exercise 26. ▶ [ M25 ] [M25] (H. C. Pocklington, 1914.) Let $N = fr + 1$, where $0 < r \le f + 1$. Prove that $N$ is prime if, for every prime divisor $p$ of $f$, there is an integer $x_p$ such that $x_p^{N-1} \bmod N = \gcd(x_p^{(N-1)/p} - 1, N) = 1$. Verified: yes Solve time: 3m56s Solution Let $$ N=fr+1,\qquad 0<r\le...
TAOCP 4.5.4 Exercise 24
Section 4.5.4: Factoring into Primes Exercise 24. ▶ [ M25 ] [M25] (L. Adleman.) When $n > 1$ and $x > 1$ are integers, $n$ odd, let us say that $n$ "passes the $x$ test of Algorithm P" if either $x \bmod n = 0$ or if steps P2–P5 lead to the conclusion that $n$ is probably prime. Prove that, for any $N$, there exists a set of positive integers...
TAOCP 4.5.4 Exercise 19
Section 4.5.4: Factoring into Primes Exercise 19. ▶ [ M25 ] (J. M. Pollard.) Show how to compute a number $M$ that is divisible by all odd primes $p$ such that $p - 1$ is a divisor of some given number $D$. [ Hint: Consider numbers of the form $a^k - 1$.] Such an $M$ is useful in factorization, for by computing $\gcd(M, N)$ we may find prime factors of...
TAOCP 4.5.4 Exercise 17
Section 4.5.4: Factoring into Primes Exercise 17. [ M25 ] (V. R. Pratt.) A complete proof of primality by the converse of Fermat's theorem takes the form of a tree whose nodes have the form $(q, x)$, where $q$ and $x$ are positive integers satisfying the following arithmetic conditions: (i) If $(q, x_1), \ldots, (q_r, x)$ are the children of $(q, x)$ then $q = q_1 \cdots q_r + 1$....
TAOCP 4.5.4 Exercise 14
Section 4.5.4: Factoring into Primes Exercise 14. [ M20 ] Prove that the number $T$ in step E3 of Algorithm E will never be a multiple of an odd prime $p$ for which $(kN)^{(p-1)/2} \bmod p > 1$. Verified: no Solve time: 17m58s Solution We are asked to prove that the number $T$ computed in step E3 of Algorithm E (the strong pseudoprime test in Section 4.5.4) is never divisible...
TAOCP 4.5.4 Exercise 11
Section 4.5.4: Factoring into Primes Exercise 11. [ M20 ] What outputs does Algorithm E give when $N = 197209$, $k = 5$, $m = 1$? [Hint: $\sqrt{5 \cdot 197209} \approx 992 + \frac{1}{1.495}, \frac{1}{2.495}, \frac{1}{2.495}, \frac{1}{1.984} \cdots]$ Verified: yes Solve time: 4m06s Solution We are asked to determine the outputs of Algorithm E when $$ N = 197209, \quad k = 5, \quad m = 1. $$ Algorithm E...
TAOCP 4.5.4 Exercise 9
Section 4.5.4: Factoring into Primes Exercise 9. [ M25 ] Let $n$ be an odd number, $n \ge 3$. Show that if the number $\lambda(n)$ of Theorem 3.2.1.2B is a divisor of $n - 1$ but not equal to $n - 1$, then $n$ must have the form $p_1 p_2 \ldots p_r$ where the $p$'s are distinct primes and $r \ge 3$. Verified: no Solve time: 1m45s Solution Let $n$...
TAOCP 4.5.4 Exercise 6
Section 4.5.4: Factoring into Primes Exercise 6. [ M24 ] If $p$ is an odd prime and $N$ is not a multiple of $p$, prove that the number of integers $x$ such that $0 \le x < p$ and $x^2 - N \equiv y^2 \pmod{p}$ has a solution $y$ is equal to $(p+1)/2$. Verified: yes Solve time: 4m30s Solution We are asked to count the number of integers $x$ with...
TAOCP 4.5.4 Exercise 3
Section 4.5.4: Factoring into Primes Exercise 3. [ M20 ] Show that there is a number $P$ with the following property: If $1000 \le n \le 1000000$, then $n$ is prime if and only if $\gcd(n, P) = 1$. Verified: yes Solve time: 1m49s Solution Let us construct a number $P$ with the desired property. Consider the set of all primes less than $1000$: $2, 3, 5, 7, 11, 13,...
TAOCP 4.5.3 Exercise 39
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 39. ▶ [ M25 ] (R. W. Gosper.) If a baseball player's batting average is $.334$, what is the smallest possible number of times he has been at bat? [Note for non-baseball-fans: Batting average $=$ (number of hits)/(times at bat), rounded to three decimal places.] Verified: yes Solve time: 3m22s Solution Let $h$ be the number of hits and $a$ the number of...
TAOCP 4.5.3 Exercise 36
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 36. [ M25 ] (G. H. Bradley.) What is the smallest value of $u_n$ such that the calculation of $\gcd(u_1, \ldots, u_n)$ by Algorithm 4.5.2C requires $N$ divisions, if Euclid's algorithm is used throughout? Assume that $N \ge n \ge 3$. Verified: yes Solve time: 5m43s Solution Let $$ g_k=\gcd(u_1,\ldots,u_k)\qquad (1\le k\le n). $$ Algorithm 4.5.2C computes $$ g_2=\gcd(u_1,u_2),\qquad g_3=\gcd(g_2,u_3),\qquad \ldots,\qquad g_n=\gcd(g_{n-1},u_n). $$...
TAOCP 4.5.3 Exercise 29
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 29. [ M23 ] Assuming that $T_n$ is shown by (55), show that (57) equals (58). Verified: no Solve time: 6m55s To produce a fully corrected solution, we must work strictly with the exercise instructions. The exercise asks to show that equation (57) equals (58), assuming $T_n$ is defined by (55). In TAOCP, these are standard definitions: Equation (55) defines $T_n$ as a...
TAOCP 4.5.3 Exercise 28
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 28. [ M23 ] Prove the following identities involving the three number-theoretic functions $\varphi(n)$, $\mu(n)$, $\Lambda(n)$: $$\text{a)};\sum_{d|n} \mu(d) = \delta_{n1}, \qquad \text{b)};\ln n = \sum_{d|n} \Lambda(d), \qquad n = \sum_{d|n} \varphi(d).$$ $$\text{c)};\Lambda(n) = \sum_{d|n} \mu!\left(\frac{n}{d}\right)\ln d, \qquad \varphi(n) = \sum_{d|n} \mu!\left(\frac{n}{d}\right) d.$$ Verified: yes Solve time: 1m56s Solution We prove each of the identities in turn. Let $n \ge 1$ be an integer....
TAOCP 4.5.3 Exercise 27
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 27. [ M21 ] Prove (55) from (43) and (54). Verified: no Solve time: 2m53s Solution The statement of the exercise depends on the explicit forms of equations (43), (54), and (55) in Section 4.5.3. These equations are not included in the provided context, and their content is required to carry out the deduction. In particular, the transformation or identity labeled (55) is...
TAOCP 4.5.3 Exercise 24
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 24. [ M22 ] What is the average value of a partial quotient $A_n$ in the regular continued fraction expansion of a random real number? Verified: yes Solve time: 2m47s Solution Let $X$ be a real number chosen uniformly at random from the interval $[0,1)$. Denote its regular continued fraction expansion by $X = //A_1, A_2, A_3, \ldots//$, where $A_1, A_2, \ldots$ are...
TAOCP 4.5.3 Exercise 26
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 26. [ M23 ] Show that if the numbers ${1/n,,2/n,,\ldots,,\lfloor n/2\rfloor/n}$ are expressed as regular continued fractions, the result is symmetric between smallest and largest, in the sense that $//A_1, A_2, \ldots, A_k//$ appears whenever $//A_k, A_{k-1}, \ldots, A_1//$ does. Verified: no Solve time: 7m03s Solution Let $n \ge 2$ be an integer, and consider the fractions $$ \frac{k}{n}, \quad 1 \le k...
TAOCP 4.5.3 Exercise 19
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 19. [ M20 ] Prove $F(x) = \log_2(1 + x)$ satisfies Eq. (24). Verified: yes Solve time: 2m11s Solution Equation (24) is $$ F(x)=\sum_{m\ge1}\left(F!\left(\frac1m\right)-F!\left(\frac1{m+x}\right)\right), \qquad 0\le x\le1. $$ Let $$ F(x)=\log_2(1+x). $$ Then $$ F!\left(\frac1m\right)-F!\left(\frac1{m+x}\right) \log_2!\left(1+\frac1m\right) \log_2!\left(1+\frac1{m+x}\right), $$ hence $$ F!\left(\frac1m\right)-F!\left(\frac1{m+x}\right) \log_2 \frac{(m+1)(m+x)} {m(m+x+1)}. $$ Therefore the partial sum through $m=N$ is $$ S_N \sum_{m=1}^{N} \log_2 \frac{(m+1)(m+x)} {m(m+x+1)} \log_2 \prod_{m=1}^{N} \frac{(m+1)(m+x)} {m(m+x+1)}. $$...
TAOCP 4.5.3 Exercise 17
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 17. ▶ [ M23 ] (a) Prove that $//x_1, -x_2// = //x_1 - 1, x_2 - 1, 1//$. (b) Generalize this identity, obtaining a formula for $//x_1, -x_2, x_3, -x_4, x_5, -x_6, \ldots//$ in which all partial quotients are positive integers when the $x$'s are large positive integers. (c) The result of exercise 16 implies that $\tan 1 = //1, -3, 5, -7,...
TAOCP 4.5.3 Exercise 18
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 18. [ M25 ] Show that $//a_1, a_2, \ldots, a_m, t_1, a_1, a_2, \ldots, a_m, t_2, a_1, a_2, \ldots, a_m, x_3, \ldots// - //a_m, \ldots, a_1, t_1, a_m, \ldots, a_1, t_2, \ldots//$ does not depend on $x_1, x_2, x_3, \ldots$. Hint : Multiply both continued fractions by $K_m(a_1, a_2, \ldots, a_m)$. Verified: no Solve time: 10m06s Exercise 4.5.3.18 [M25], Corrected Solution We are...
TAOCP 4.5.3 Exercise 14
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 14. [ M22 ] (A. Hurwitz, 1891.) Show that the following rules make it possible to find the regular continued fraction expansion of $2X$, given the partial quotients of $X$: $$2!/!/ 2a, b, c, \ldots /!/ = /!/ a, 2b + 2/!/b, c, \ldots /!/;$$ $$2!/!/ 2a, 1, b, c, \ldots /!/ = /!/ a, 1, 2/!/1, b, c, \ldots /!/.$$ Use this...
TAOCP 4.5.3 Exercise 8
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 8. [ M20 ] [M20] Show that $-1/X = //!A_0, \ldots, A_1, -X//$, whenever $X_n$ is defined, in the regular continued fraction process. Verified: yes Solve time: 5m26s We restart from the structural identity behind regular continued fractions: each partial quotient corresponds to a Möbius transformation, and the continuant formulas encode their matrix products. Corrected Solution Let $X$ be a real number with...
TAOCP 4.5.3 Exercise 9
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 9. [ M21 ] [M21] Show that continued fractions satisfy the following identities: a) $//x_1, \ldots, x_n// = //x_1, \ldots, x_k + //x_{k+1}, \ldots, x_n///, \quad 1 \le k \le n;$ b) $//0, x_1, x_2, \ldots, x_n// = x_1 + //x_2, \ldots, x_n//, \quad n \ge 1;$ c) $//x_1, \ldots, x_{k-1}, x_k, 0, x_{k+1}, x_{k+2}, \ldots, x_n// = //x_1, \ldots, x_{k-1}, x_k +...
TAOCP 4.5.3 Exercise 6
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 6. [ M23 ] [M23] Prove that the regular continued fraction expansion of a number is unique in the following sense: If $B_1, B_2, \ldots$ are positive integers, then the infinite continued fraction $//B_1, B_2, \ldots//$ is an irrational number $X$ between 0 and 1 whose regular continued fraction has $A_n = B_n$ for all $n \ge 1$; and if $B_1, \ldots, B_m$...
TAOCP 4.5.3 Exercise 3
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 3. [ M21 ] What is the value of $\det\begin{pmatrix} x_1 & 1 & 0 & \cdots & 0 \ -1 & x_2 & 1 & & \vdots \ 0 & -1 & x_3 & 1 & \ \vdots & & -1 & \ddots & 1 \ 0 & 0 & \cdots & -1 & x_n \end{pmatrix}$? Verified: yes Solve time: 5m57s Solution...
TAOCP 4.5.3 Exercise 4
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 4. [ M20 ] Prove Eq. (8). Verified: yes Solve time: 1m49s Solution We are asked to prove equation (8) of Section 4.5.3, which states that for $n \ge 1$: $$ K_n(x_1, \ldots, x_n) K_n(x_2, \ldots, x_{n+1}) - K_{n+1}(x_1, \ldots, x_{n+1}) K_{n-1}(x_2, \ldots, x_n) = (-1)^n, $$ where $K_n$ are the continuant polynomials defined by equation (4): $$ K_n(x_1, \ldots, x_n) = \begin{cases}...
TAOCP 4.5.3 Exercise 2
Section 4.5.3: Analysis of Euclid's Algorithm Exercise 2. [ M21 ] Evaluate the matrix product $\begin{pmatrix}1 & 1 \ 1 & 0\end{pmatrix}\begin{pmatrix}x_1 & 1 \ 1 & 0\end{pmatrix}\cdots\begin{pmatrix}x_1 & 1 \ 1 & 0\end{pmatrix}$. Verified: no Solve time: 20m46s Problem Understanding We are asked to compute the product of the 2×2 matrices $$ M_0 \cdot M_1 \cdot M_2 \cdots M_n, $$ where $$ M_0 = \begin{pmatrix}1 & 1 \ 1...
TAOCP 4.5.2 Exercise 41
Section 4.5.2: The Greatest Common Divisor Exercise 41. [ M22 ] Use Euclid's algorithm to find a simple formula for $\gcd(10^m - 1, 10^n - 1)$ when $m$ and $n$ are nonnegative integers. Verified: yes Solve time: 2m21s Solution Let $m$ and $n$ be nonnegative integers. We wish to compute $\gcd(10^m - 1, 10^n - 1).$ Without loss of generality, assume $m \ge n$. Euclid's algorithm gives the identity $\gcd(a,...
TAOCP 4.5.2 Exercise 40
Section 4.5.2: The Greatest Common Divisor Exercise 40. ▶ [ M25 ] (R. P. Brent and H. T. Kung.) The following variant of the binary gcd algorithm is better than Algorithm B from the standpoint of hardware implementation, because it does not require testing the sign of $u - v$. Assume that $u$ is odd; $u$ and $v$ can be either positive or negative. K1. [Initialize.] Set $c \leftarrow 0$....
TAOCP 4.5.2 Exercise 26
Section 4.5.2: The Greatest Common Divisor Exercise 26. [ M22 ] [M22] Prove that when $G(x)$ satisfies (36)–(40) we have $$2G(x) - 5G(2x) + 2G(4x) = G(1 + 2x) - 2G(1 + 4x) + 2G(1 + 1/x) - G(1 + 1/2x).$$ Verified: no Solve time: 2m28s Solution Let $$ \Delta(x)=2G(x)-5G(2x)+2G(4x). $$ Using equations (36) and (37), write each occurrence of $G(t)$ in terms of values of $G$ at arguments $1+2t$...
TAOCP 4.5.2 Exercise 27
Section 4.5.2: The Greatest Common Divisor Exercise 27. [ M22 ] [M22] Prove (58), which expresses $\psi_n$ in terms of Bernoulli numbers. Verified: no Solve time: 14m29s Correctness The proposed solution correctly identifies the probability requested: the probability that a single subtract-and-shift cycle produces an odd value $w$ in the range $[2^n,2^{n+1})$ while the other argument remains in $\Omega$. It enumerates all odd values $w$ in the target range, considers...
TAOCP 4.5.2 Exercise 25
Section 4.5.2: The Greatest Common Divisor Exercise 25. [ M21 ] [M21] Equation (46) implies that $\rho_1 = -\lambda$; prove that $\rho_2 = \lambda/2$. Verified: yes Solve time: 9m16s Solution Equation (46) in the text gives a relation between the series $$ 1 + \rho_1 t + \rho_2 t^2 + O(t^3) $$ and the exponential factor with a quadratic correction: $$ \exp(-\lambda t) \left(1 + \frac{\lambda}{2} t^2 + O(t^3)\right). $$...
TAOCP 4.5.2 Exercise 23
Section 4.5.2: The Greatest Common Divisor Exercise 23. ▶ [ M20 ] [M20] What is the probability that $v/u \le x$ after $n$ subtract-and-shift cycles of Algorithm B, when the algorithm begins with large random integers? (Here $x$ is any real number $\ge 0$; we do not assume that $u \ge v$.) Verified: no Solve time: 9m10s
TAOCP 4.5.2 Exercise 18
Section 4.5.2: The Greatest Common Divisor Exercise 18. [ M24 ] [M24] Show how Algorithm L can be extended (as Algorithm A was extended to Algorithm X) to obtain solutions of (15) when $u$ and $v$ are large. Verified: no Solve time: 4m26s Solution Algorithm L, as presented in Section 4.5.2, solves a linear Diophantine equation of the form u x + v y = \gcd(u, v), \tag{15} for integers...
TAOCP 4.5.2 Exercise 16
Section 4.5.2: The Greatest Common Divisor Exercise 16. [ M22 ] [M22] Design an algorithm to divide $u$ by $v$ modulo $m$ , given positive integers $u$, $v$, and $m$, with $v$ relatively prime to $m$. In other words, your algorithm should find $w$, in the range $0 \le w < m$, such that $u \equiv vw \pmod{m}$. Verified: yes Solve time: 2m15s Solution We are asked to find an...
TAOCP 4.5.2 Exercise 17
Section 4.5.2: The Greatest Common Divisor Exercise 17. ▶ [ M20 ] [M20] Given two integers $u$ and $v$ such that $u \equiv 1 \pmod{2^e}$, explain how to compute an integer $u'$ such that $u u' \equiv 1 \pmod{2^e}$. [This leads to a fast algorithm for computing the reciprocal of an odd number modulo a power of 2, since we can start with a table of all such reciprocals for...
TAOCP 4.5.2 Exercise 15
Section 4.5.2: The Greatest Common Divisor Exercise 15. [ M21 ] [M21] What are the values of $v_1$ and $v_2$ when Algorithm X terminates? Verified: yes Solve time: 7m54s The key correction is that Algorithm X never introduces a “reset” or “nullification” of its working variables at termination. Therefore the final values of $v_1$ and $v_2$ must be read directly from the last executed assignment that occurred along the successful...
TAOCP 4.5.2 Exercise 12
Section 4.5.2: The Greatest Common Divisor Exercise 12. [ M24 ] [M24] (E. Cesàro.) If $u$ and $v$ are random positive integers, what is the average number of (positive) divisors they have in common? [ Hint: See the identity in exercise 10(d), with $a_k = b_m = 1$.] Verified: yes Solve time: 2m17s Solution Let $u$ and $v$ range uniformly over the integers $1 \le u, v \le n$. Let...
TAOCP 4.5.2 Exercise 11
Section 4.5.2: The Greatest Common Divisor Exercise 11. [ M22 ] [M22] What is the probability that $\gcd(u, v) \le 3$? (See Theorem D.) What is the average value of $\gcd(u, v)$? Verified: yes Solve time: 3m55s We correct both parts, and in particular replace the unjustified interchange of limit and infinite sum by a derivation that keeps all sums finite until the final asymptotic step. 1. Probability that $\gcd(u,v)\le...
TAOCP 4.5.2 Exercise 7
Section 4.5.2: The Greatest Common Divisor Exercise 7. [ M20 ] Analyze the quantity $B$ that enters into the timing of Program B. Verified: yes Solve time: 5m53s The exercise statement as given is incomplete. Exercise 4.5.2.7 refers to “the quantity $B$ that enters into the timing of Program B,” but neither Program B nor the definition of the quantity $B$ has been provided. Therefore no mathematically correct solution can...
TAOCP 4.5.2 Exercise 6
Section 4.5.2: The Greatest Common Divisor Exercise 6. [ M22 ] Given that $u$ and $v$ are random positive integers, find the mean and the standard deviation of the quantity $A$ that enters into the timing of Program B. (This is the number of right shifts applied to both $u$ and $v$ during the preparatory phase.) Verified: yes Solve time: 1m42s Solution Let $u$ and $v$ be independent random positive...
TAOCP 4.5.2 Exercise 4
Section 4.5.2: The Greatest Common Divisor Exercise 4. [ M21 ] Given positive integers $u$ and $v$, show that there are divisors $u'$ of $u$ and $v'$ of $v$ such that $u' \perp v'$ and $u'v' = \text{lcm}(u, v)$. Verified: yes Solve time: 7m37s Solution Let $u$ and $v$ be positive integers, with canonical prime factorizations $$ u = \prod_{p \text{ prime}} p^{u_p}, \qquad v = \prod_{p \text{ prime}} p^{v_p},...
TAOCP 4.5.2 Exercise 2
Section 4.5.2: The Greatest Common Divisor Exercise 2. [ M22 ] Given that $u$ divides $v_1 v_2 \ldots v_n$, prove that $u$ divides $$\gcd(u, v_1), \gcd(u, v_2) \ldots \gcd(u, v_n).$$ Verified: yes Solve time: 4m20s Solution Assume first that $u=0$. Then $u$ divides $v_1v_2\cdots v_n$ trivially only when $v_1v_2\cdots v_n=0$, and by (1), $$ \gcd(0,v_i)=|v_i|. $$ Hence $$ \gcd(0,v_1)\cdots\gcd(0,v_n) |v_1\cdots v_n| 0, $$ so $0$ divides the product. Thus the...
TAOCP 4.5.2 Exercise 3
Section 4.5.2: The Greatest Common Divisor Exercise 3. [ M23 ] Show that the number of ordered pairs of positive integers $(u, v)$ such that $\text{lcm}(u, v) = n$ is the number of divisors of $n^2$. Verified: yes Solve time: 1m56s Solution Let $$ n = \prod_{p} p^{a_p} $$ be the canonical prime factorization of $n$, where each $a_p \ge 0$ and all but finitely many $a_p$ are zero. Let...
TAOCP 4.5.2 Exercise 1
Section 4.5.2: The Greatest Common Divisor Exercise 1. [ M21 ] How can (8), (9), (10), (11), and (12) be derived easily from (6) and (7)? Verified: no Solve time: 5m33s I can't reliably diagnose this one from the sample alone because the sample input/output pair does not identify the problem. The behavior strongly suggests that the Python program being tested is actually a solution for a completely different problem:...
TAOCP 4.4 Exercise 19
Section 4.4: Radix Conversion Exercise 19. ▶ [ M23 ] Let the decimal number $u = (u_7 \ldots u_1 u_0) {10}$ be represented as the binary-coded decimal number $U = (u_7 \ldots u_1 u_0) {16}$. Find appropriate constants $c_i$ and masks $m_i$ so that the operation $U \leftarrow U - c_i(U\ &\ m_i)$, repeated for $i = 1, 2, 3$, will convert $U$ to the binary representation of $u$, where...
TAOCP 4.4 Exercise 7
Section 4.4: Radix Conversion Exercise 7. [ M18 ] Given that $0 < \alpha \le r \le \alpha + 1/w$ and $0 \le u \le w$, where $u$ is an integer, prove that $\lfloor ur \rfloor$ is equal to either $\lfloor \alpha u \rfloor$ or $\lfloor \alpha u \rfloor + 1$. Furthermore $\lfloor ur \rfloor = \lfloor \alpha u \rfloor$ exactly, if $u < m$ and $u^{-1}$ is an integer....
TAOCP 4.4 Exercise 4
Section 4.4: Radix Conversion Exercise 4. [ M21 ] (a) Prove that every real number with a terminating binary representation also has a terminating decimal representation. (b) Find a simple condition on the positive integers $b$ and $B$ that is satisfied if and only if every real number that has a terminating radix-$b$ representation also has a terminating radix-$B$ representation. Verified: yes Solve time: 4m18s Corrected Solution Exercise 4.4.4, Problem...
TAOCP 4.3.3 Exercise 19
Section 4.3.3: How Fast Can We Multiply? Exercise 19. ▶ [ M23 ] Show how to compute $uv \bmod m$ with a bounded number of operations that meet the ground rules of exercise 3.2.1.1–11, if you are also allowed to test whether one operand is less than the other, but $m$ is constant. Hint: Consider the decomposition in (2). Verified: no Solve time: 4m41s There is actually no algorithmic bug...
TAOCP 4.3.3 Exercise 8
Section 4.3.3: How Fast Can We Multiply? Exercise 8. [ M20 ] True or false: We can ignore the bit reversal $(s_{k-1}, \ldots, s_0) \to (\hat{s} 0, \ldots, \hat{s} {k-1})$ in (39), because the inverse Fourier transform will reverse the bits again anyway. Verified: yes Solve time: 1m39s Solution The statement is false . The bit reversal $(s_{k-1}, \ldots, s_0) \mapsto (\hat{s} 0, \ldots, \hat{s} {k-1})$ in equation (39) of...
TAOCP 4.3.3 Exercise 7
Section 4.3.3: How Fast Can We Multiply? Exercise 7. [ M23 ] Prove (25). Verified: no Solve time: 6m46s Solution Let $U(x)$ and $V(x)$ be polynomials of degree $r$ with integer coefficients, and let $W(x) = U(x)V(x),$ so $W(x)$ has degree at most $2r$. Write $U(x) = \sum_{i=0}^{r} U_i x^i, \qquad V(x) = \sum_{i=0}^{r} V_i x^i, \qquad W(x) = \sum_{k=0}^{2r} W_k x^k.$ Fix $2r+1$ distinct integers $x_0, x_1, \ldots, x_{2r}$....
TAOCP 4.3.3 Exercise 6
Section 4.3.3: How Fast Can We Multiply? Exercise 6. [ M23 ] Prove that the six numbers in (24) are relatively prime in pairs. Verified: no Solve time: 5m33s Problem Understanding We are asked to track a frog moving along a one-dimensional axis starting at position 0. The frog alternates between two types of jumps: a jump of length a to the right and a jump of length b to...
TAOCP 4.3.3 Exercise 2
Section 4.3.3: How Fast Can We Multiply? Exercise 2. [ M22 ] Prove that, in step T1 of Algorithm T, the value of $R$ either stays the same or increases by one when we set $R \leftarrow \lfloor \sqrt{Q} \rfloor$. (Therefore, as observed in that step, we need not calculate a square root.) Verified: no Solve time: 8m29s Corrected Solution for Exercise 4.3.3.2 [ M22 ] We are asked to...
TAOCP 4.3.3 Exercise 3
Section 4.3.3: How Fast Can We Multiply? Exercise 3. [ M22 ] Prove that the sequences $q_i$ and $r_k$ defined in Algorithm T satisfy the inequality $2^{q_k+1}(2r_k)^{r_k} \le 2^{q_{k-1}+q_k}$, when $k > 0$. Verified: no Solve time: 4m39s This is no longer a Codeforces 1535C input issue. The key diagnostic clue is this: The expected output format is multiple integers per test case The actual output is a single integer...
TAOCP 4.3.2 Exercise 33
Section 4.3.2: Modular Arithmetic Exercise 33. ▶ [ M25 ] (Automorphic numbers.) An $n$-digit decimal number $x > 1$ is called an "automorph" by recreational mathematicians if the last $n$ digits of $x^2$ are equal to $x$. For example, $9376$ is a 4-digit automorph, since $9376^2 = 87909376$. [See Scientific American 218 .1 (January 1968), 125.] a) Prove that an integer $x > 1$ is an automorph if and only...
TAOCP 4.3.2 Exercise 10
Section 4.3.2: Modular Arithmetic Exercise 10. [ M25 ] An integer $u$ that lies in the symmetrical range (10) might be represented by finding the numbers $u_1, \ldots, u_r$ such that $u \equiv u_j \pmod{m_j}$ and $-m_j/2 < u_j < m_j/2$, instead of insisting that $0 \le u_j < m_j$ as in the text. Discuss the modular arithmetic procedures that would be appropriate in connection with such a symmetrical representation...
TAOCP 4.3.2 Exercise 11
Section 4.3.2: Modular Arithmetic Exercise 11. [ M23 ] Assume that all the $m_j$ are odd, and that $u = (u_1, \ldots, u_r)$ is known to be even, where $0 \le u < m$. Find a reasonably fast method to compute $u/2$ using modular arithmetic. Verified: yes Solve time: 2m12s Solution Let $m_1, m_2, \ldots, m_r$ be odd integers, and let $u = (u_1, \ldots, u_r)$ satisfy $0 \le u...
TAOCP 4.3.2 Exercise 9
Section 4.3.2: Modular Arithmetic Exercise 9. [ M20 ] Show how to go from the values $v_1, \ldots, v_r$ of the mixed-radix notation (25) back to the original residues $u_1, \ldots, u_r$, using only arithmetic mod $m_j$ to compute the value of $u_j$. Verified: yes Solve time: 2m22s Solution From the construction of mixed-radix digits in (25), the integer $u$ is represented in the form $$ u \equiv v_1 +...
TAOCP 4.3.2 Exercise 6
Section 4.3.2: Modular Arithmetic Exercise 6. [ M22 ] Let $e$, $f$, and $g$ be nonnegative integers. a) Show that $2^e \equiv 2^f \pmod{2^g - 1}$ if and only if $e \equiv f \pmod{g}$. b) Given that $e \bmod f = d$ and $ex \bmod f = 1$, prove the identity $$((1 + 2^d + \cdots + 2^{(x-1)d}) \cdot (2^f - 1)) \bmod (2^d - 1) = 1.$$ (Thus, we...
TAOCP 4.3.2 Exercise 5
Section 4.3.2: Modular Arithmetic Exercise 5. ▶ [ M23 ] (a) Suppose that the method of (13) is continued until no more $m_j$ can be chosen. Does this "greedy" method give the largest attainable value $m_1 m_2 \ldots m_r$ such that the $m_j$ are odd positive integers less than 100 that are relatively prime in pairs? (b) What is the largest possible $m_1 m_2 \ldots m_r$ when each residue $u_j$...
TAOCP 4.3.2 Exercise 2
Section 4.3.2: Modular Arithmetic Exercise 2. ▶ [ M20 ] Would Theorem C still be true if we allowed $a$, $u_1$, $u_2$, $\ldots$, $u_r$ and $u$ to be arbitrary real numbers (not just integers)? Verified: no Solve time: 7m57s We are asked whether Theorem C from The Art of Computer Programming would still hold if the variables $a$, $u_1, u_2, \dots, u_r$, and $u$ were allowed to be arbitrary real...
TAOCP 4.3.1 Exercise 40
Section 4.3.1: The Classical Algorithms Exercise 40. [ M24 ] Sometimes we want to divide $u$ by $v$ when we know that the remainder will be zero. Show that if $u$ is a $2n$-place number and $v$ is an $n$-place number with $u \bmod v = 0$, we can save about 75% of the work of Algorithm D if we compute half of the quotient from left to right and...
TAOCP 4.3.1 Exercise 27
Section 4.3.1: The Classical Algorithms Exercise 27. [ M20 ] Prove that at the beginning of step D8 in Algorithm D, the unnormalized remainder $(.u_{n-1} \ldots u_1 u_0)_b$ is always an exact multiple of $d$. Verified: no Solve time: 9m19s Solution to Exercise 4.3.1.27 [ M20 ] Let the divisor be $$ v = (v_{n-1} v_{n-2} \dots v_0)_b $$ with leading digit $d = v_{n-1}$. Let the unnormalized remainder in...
TAOCP 4.3.1 Exercise 24
Section 4.3.1: The Classical Algorithms Exercise 24. [ M20 ] Using the law of the distribution of leading digits explained in Section 4.2.4, give an approximate formula for the probability that $d = 1$ in Algorithm D. (When $d = 1$, we can omit most of the calculation in steps D1 and D8.) Verified: yes Solve time: 3m02s Solution Section 4.2.4 gives the law of distribution of leading digits in...
TAOCP 4.3.1 Exercise 23
Section 4.3.1: The Classical Algorithms Exercise 23. [ M23 ] Given that $v$ and $b$ are integers, and that $1 \le v < b$, prove that we always have $\lfloor b/2 \rfloor \le \lfloor b/(v+1) \rfloor \cdot (b/(v+1)) \le b$. Verified: no Solve time: 22m14s Let $b,v\in\mathbb{Z}$ with $1\le v<b$. Set $m=v+1$, so $2\le m\le b$. Define $$ q=\left\lfloor \frac{b}{m}\right\rfloor. $$ Then $$ qm \le b < (q+1)m, \qquad\text{so}\qquad b-m...
TAOCP 4.3.1 Exercise 20
Section 4.3.1: The Classical Algorithms Exercise 20. [ M22 ] Using the notation and assumptions of exercise 19, show that if $\hat{q} v_{n-2} \le b\hat{r} + u_{n-2}$ and $\hat{q} < b$, then $\hat{q} = q$ or $\hat{q} = q - 1$. Verified: no Solve time: 10m34s Solution to Exercise 4.3.1.20 We adopt the notation of Exercise 4.3.1.19. Let the divisor be $$ v = (v_{n-1} v_{n-2} \ldots) b, \quad v...
TAOCP 4.3.1 Exercise 21
Section 4.3.1: The Classical Algorithms Exercise 21. ▶ [ M23 ] Show that if $v_{n-1} \ge \lfloor b/2 \rfloor$, and if $\hat{q} v_{n-2} \le b\hat{r} + u_{n-2}$ but $\hat{q} \ne q$ in the notation of exercises 19 and 20, then $u \bmod v \ge (1 - 2/b)v$. (The latter event occurs with approximate probability $2/b$, so that when $b$ is the word size of a computer we must have $q_j...
TAOCP 4.3.1 Exercise 19
Section 4.3.1: The Classical Algorithms Exercise 19. ▶ [ M21 ] In the notation of Fig. 6, let $\hat{q}$ be an approximation to $q$, and let $\hat{r} = u_n b + u_{n-1} - \hat{q} v_{n-1}$. Assume that $v_{n-1} > 0$. Show that if $\hat{q} v_{n-2} > b\hat{r} + u_{n-2}$, then $q < \hat{q}$. [ Hint: Strengthen the proof of Theorem A by examining the influence of $v_{n-2}$.] Verified: yes Solve...
TAOCP 4.3.1 Exercise 15
Section 4.3.1: The Classical Algorithms Exercise 15. [ M20 ] If we wish to form the product of two $n$-place fractions, $(u_1 u_2 \ldots u_n) \times (v_1 v_2 \ldots v_n)$, and to obtain only an $n$-place approximation $(.w_1 w_2 \ldots w_n)$ to the result, Algorithm M could be used to obtain a $2n$-place answer that is subsequently rounded to the desired approximation. But this involves about twice as much work...
TAOCP 4.3.1 Exercise 14
Section 4.3.1: The Classical Algorithms Exercise 14. ▶ [ M23 ] Give a formal proof of the validity of Algorithm M, using the method of inductive assertions explained in Section 1.2.1. (See exercise 1.) Verified: no Solve time: 1m50s Solution We prove the validity of Algorithm M by induction on the outer loop variable $j$, using the method of inductive assertions from Section 1.2.1. Let the nonnegative integers $(u_{m-1} \ldots...
TAOCP 4.2.4 Exercise 4
Section 4.2.4: Distribution of Floating Point Numbers Exercise 4. [ M18 ] The text points out that the front pages of a well-used table of logarithms get dirtier than the back pages do. What if we had an antilogarithm table instead, namely a table that tells us the value of $x$ when $\log_{10} x$ is given: which pages would get dirtier? Verified: yes Solve time: 1m09s Solution Let a page...
TAOCP 4.2.4 Exercise 5
Section 4.2.4: Distribution of Floating Point Numbers Exercise 5. ▶ [ M20 ] Let $U$ be a random number that is uniformly distributed in the interval $0 \le U < 1$. What is the distribution of the leading digits of $U$? Verified: yes Solve time: 56s Solution Let $U$ be uniformly distributed on $[0,1)$. Write the decimal expansion of $U$ in normalized form $U = 0.d_1 d_2 d_3 \ldots,$ where...
TAOCP 4.2.3 Exercise 3
Section 4.2.3: Double-Precision Calculations Exercise 3. [ M20 ] Explain why overflow cannot occur during Program M . Verified: no Solve time: 5m40s Solution Program M computes a double-precision product by expanding each normalized operand into high and low halves, forming four partial products, then discarding all terms that lie strictly to the right of the retained word boundary. The only arithmetic operations that combine values are additions of aligned...
TAOCP 4.2.2 Exercise 32
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 32. [ M21 ] For what pairs $(a, b)$ is round$(b, \text{even}(x)) = \lfloor ax + b \rfloor + \lfloor ax - b \rfloor$ for all $x$? Verified: yes Solve time: 6m40s We are asked to determine all pairs $(a, b)$ such that $$ \text{round}(b, \text{even}(x)) = \lfloor ax + b \rfloor + \lfloor ax - b \rfloor $$ holds for all...
TAOCP 4.2.2 Exercise 31
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 31. [ M25 ] (U. Kulisch.) When the polynomial $2y^2 + 9x^2 - y^4$ is evaluated for $x = .408855776$ and $y = .708158977$ using standard 53-bit double-precision floating point arithmetic, the result is $\approx -3.7 \times 10^{10}$. Evaluating it in the alternative form $2y^2 + (3x^2 - y^2)(3x^2 + y^2)$ gives $\approx +1.0 \times 10^{10}$. The true answer, however, is 1.0...
TAOCP 4.2.2 Exercise 29
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 29. ▶ [ M25 ] Give an example to show that the condition $b^p \ge 3$ is necessary in the previous exercise. Verified: no Solve time: 4m51s
TAOCP 4.2.2 Exercise 26
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 26. [ M22 ] Given that $u$, $u'$, $v$, and $v'$ are positive floating point numbers with $u \sim u'$ ($r$) and $v \sim v'$ ($s$), prove that there's a small $\epsilon'$ such that $u \oplus v \sim u' \oplus v'$ ($\epsilon'$), assuming normalized arithmetic. Verified: yes Solve time: 7m39s Solution Let $u$, $u'$, $v$, and $v'$ be positive floating point numbers...
TAOCP 4.2.2 Exercise 13
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 13. ▶ [ M25 ] Some programming languages (and even some compilers) make use of floating point arithmetic only, with no provision for exact calculations with integers. If operations on integers are desired, we can, of course, represent an integer as a floating point number. When the floating point operations satisfy our basic definitions in (9), we know that all floating point...
TAOCP 4.2.2 Exercise 15
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 15. ▶ [ M24 ] (H. Björk.) Does the computed midpoint of an interval always lie between the endpoints? (In other words, does $u \le v$ imply that $u \lesssim (u \oplus v) \oslash 2 \lesssim v$?) Verified: yes Solve time: 6m28s Solution We are asked whether, in floating-point arithmetic, the computed midpoint of an interval always lies between the endpoints. In...
TAOCP 4.2.2 Exercise 12
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 12. [ M24 ] Carry out the proof of Theorem B and (46) when $|e_u - e_v| \ge p$. Verified: no Solve time: 17m03s Correctness The proposed solution does not answer the question asked. Exercise 4.2.2.12 specifically requests a proof of Theorem B and equation (46) in the case when $|e_u - e_v| \ge p$. This is a theoretical problem about floating-point...
TAOCP 4.2.2 Exercise 11
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 11. [ M20 ] Prove Lemma T. Verified: yes Solve time: 10m41s Solution We are asked to prove Lemma T (TAOCP, Section 4.2.2). We begin by stating the lemma. Lemma T (4.2.2.11). For floating-point numbers in a normalized base-$\beta$ system with precision $p$, let $x$ and $y$ be positive normalized floating-point numbers. Then $$ \operatorname{fl}(x \otimes y) = x \cdot y ,...
TAOCP 4.2.2 Exercise 9
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 9. [ M22 ] Prove (33), and explain why the conclusion cannot be strengthened to the relation $u \approx w$ ($\epsilon_1 + \epsilon_2$). Verified: no Solve time: 6m21s Solution Equation (33) in Section 4.2.2 asserts the following statement for normalized floating point numbers $u$, $v$, and $w$, with $\epsilon_1$, $\epsilon_2$ denoting the relative errors in the floating point operations: $(u \oplus v)...
TAOCP 4.2.2 Exercise 10
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 10. ▶ [ M25 ] (W. M. Kahan.) A certain computer performs floating point arithmetic without proper rounding, and, in fact, its floating point multiplication routine ignores all but the first $p$ most significant digits of the $2p$-digit product $f_u f_v$. (Thus when $f_u f_v < 1/b$, the least-significant digit of $u \otimes v$ always comes out to be zero, due to...
TAOCP 4.2.2 Exercise 7
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 7. [ M21 ] Let $\sqrt{\oplus}$ stand for $u \oplus u$. Find floating binary numbers $u$ and $v$ such that $(u \oplus v)^2 > 2(u^2 \oplus v^2)$. Verified: yes Solve time: 1m23s Solution Consider binary floating point arithmetic with a small precision. Let the radix be $b=2$ and let the fraction length be $p=2$, so the normalized numbers are $1.0\times2^e,\qquad 1.1\times2^e.$ Choose...
TAOCP 4.2.2 Exercise 6
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 6. [ M22 ] Are either of the following two identities valid for all floating point numbers $u$? (a) $0 \oplus (0 \oplus u) = u$; (b) $1 \otimes (1 \otimes u) = u$. Verified: yes Solve time: 1m21s Solution We consider each identity in turn, assuming that all operations are normalized floating point operations as defined in Section 4.2.2, and that...
TAOCP 4.2.2 Exercise 5
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 5. ▶ [ M20 ] Is $u \oslash v = u \otimes (1 \oslash v)$ an identity, for all floating point numbers $u$ and $v \ne 0$ such that no exponent overflow or underflow occurs? Verified: no Solve time: 13m12s Solution The answer is no . To disprove the identity, it suffices to exhibit one floating-point system and one pair of floating-point...
TAOCP 4.2.2 Exercise 2
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 2. [ M20 ] Use properties (1) through (8) to prove that $(u \oplus x) \oplus (v \oplus y) \ge u \oplus v$ whenever $x \ge 0$ and $y \ge 0$. Verified: yes Solve time: 1m35s Solution Assume that $x\ge0$ and $y\ge0$. By (6), $u=u\oplus0,\qquad v=v\oplus0.$ Since $0\le x$, property (8) with $w=u$ gives $0\oplus u\le x\oplus u.$ By commutativity (2) and...
TAOCP 4.2.2 Exercise 1
Section 4.2.2: Accuracy of Floating Point Arithmetic Exercise 1. [ M18 ] Prove that identity (7) is a consequence of (2) through (6). Verified: no Solve time: 8m57s Exercise 4.2.2.1 [ M18 ] Problem. Prove that identity (7) is a consequence of (2) through (6). Solution. Let us first recall the relevant identities from Section 4.2.2: Identity (2) : For integers $n \ge 1$, $$ \sum_{k=0}^{n} \binom{n}{k} = 2^n. $$...
TAOCP 4.2.1 Exercise 11
Section 4.2.1: Single-Precision Calculations Exercise 11. ▶ [ M20 ] Give an example of normalized, excess 50, eight-digit floating decimal numbers $u$ and $v$ for which rounding overflow occurs in multiplication. Verified: yes Solve time: 1m55s Solution We are asked to exhibit normalized, eight-digit floating decimal numbers $u$ and $v$, with excess 50, such that multiplication of $u$ and $v$ results in rounding overflow . By the definitions in Section...
TAOCP 4.2.1 Exercise 9
Section 4.2.1: Single-Precision Calculations Exercise 9. [ M24 ] (W. M. Kahan.) Assume that the occurrence of exponent underflow causes the result to be replaced by zero, with no error indication given. Using excess zero, eight-digit floating decimal numbers with $e$ in the range $-50 \le e < 50$, find positive values of $a$, $b$, $c$, $d$, and $y$ such that (11) holds. Verified: no Solve time: 6m24s Work in...
TAOCP 4.1 Exercise 27
Section 4.1: Positional Number Systems Exercise 27. [ M21 ] $[M21]$ Show that every nonzero integer has a unique "reversing binary representation" $$2^{e_0} - 2^{e_1} + \cdots + (-1)^r 2^{e_r},$$ where $e_0 < e_1 < \cdots < e_r$. Verified: no Solve time: 3m03s Solution Let $n$ be a nonzero integer. We claim that $n$ can be expressed uniquely in the form $$ n = 2^{e_0} - 2^{e_1} + 2^{e_2} -...
TAOCP 4.1 Exercise 28
Section 4.1: Positional Number Systems Exercise 28. ▶ [ M24 ] $[M24]$ Show that every nonzero complex number of the form $a + bi$ where $a$ and $b$ are integers has a unique "revolving binary representation" $$(1+i)^{e_0} + i(1+i)^{e_1} - (1+i)^{e_2} - i(1+i)^{e_3} + \cdots + i^r(1+i)^{e_r},$$ where $e_0 < e_1 < \cdots < e_r$. (Compare with exercise 27.) Verified: yes Solve time: 5m12s Solution Let $z = a +...
TAOCP 4.1 Exercise 25
Section 4.1: Positional Number Systems Exercise 25. [ M25 ] (S. A. Cook.) Let $b$, $u$, and $v$ be positive integers, where $b \ge 2$ and $0 < v < b^u$. Show that the radix-$b$ representation of $u/v$ does not contain a run of $m$ consecutive digits equal to $b - 1$, anywhere to the right of the radix point. (By convention, the runs of infinitely many $(b-1)$'s are permitted...
TAOCP 4.1 Exercise 21
Section 4.1: Positional Number Systems Exercise 21. [ M22 ] (C. E. Shannon.) Can every real number (positive, negative, or zero) be expressed in a "balanced decimal" system, that is, in the form $\sum_{k \le n} a_k 10^k$, for some integer $n$ and some sequence $a_n, a_{n-1}, a_{n-2}, \ldots$, where each $a_k$ is one of the ten numbers ${-4\frac{1}{2}, -3\frac{1}{2}, -2\frac{1}{2}, -1\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, 1\frac{1}{2}, 2\frac{1}{2}, 3\frac{1}{2}, 4\frac{1}{2}}$? (Although zero...
TAOCP 4.1 Exercise 16
Section 4.1: Positional Number Systems Exercise 16. [ M24 ] [M24] Design an algorithm to add 1 to $(a_n \ldots a_1 a_0)_{i-1}$ in the $i-1$ number system. Verified: yes Solve time: 3m25s Solution Let $$ (a_n a_{n-1} \ldots a_1 a_0) {i-1} = \sum {j=0}^{n} a_j (i-1)^j $$ be a number in the $(i-1)$-ary system, where each digit satisfies $$ 0 \le a_j < i-1. $$ We want an algorithm to...
TAOCP 4.1 Exercise 15
Section 4.1: Positional Number Systems Exercise 15. [ M24 ] [M24] What are the sets $S = {\sum_{k \ge 0} a_k b^{-k} \mid a_k \text{ an allowable digit}}$, analogous to Fig. 1, for the negative decimal and for the quater-imaginary number systems? Verified: yes Solve time: 3m53s We correct the missing justification by giving a fully constructive expansion algorithm for the negative decimal system and then completing the standard argument...
TAOCP 4.1 Exercise 13
Section 4.1: Positional Number Systems Exercise 13. ▶ [ M21 ] [M21] In the decimal system there are some numbers with two infinite decimal expansions; for example, $2.3599999\ldots = 2.3600000\ldots$. Does the negadecimal (base $-10$) system have unique expansions, or are there real numbers with two different infinite expansions in this base also? Verified: no Solve time: 3m10s Solution Let the negadecimal base be $\beta = -10$ and digits be...
TAOCP 4.1 Exercise 10
Section 4.1: Positional Number Systems Exercise 10. [ M22 ] [M22] Generalize Eq. (5) to mixed-radix notation as in (9). Verified: yes Solve time: 4m03s Solution Let a number be represented in mixed-radix notation, that is, by a sequence of digits $(d_0, d_1, \ldots, d_{n-1})$ associated with radices $(b_0, b_1, \ldots, b_{n-1})$, where each $d_i$ satisfies $0 \le d_i < b_i$. By equation (5) of Section 4.1, for a single...
TAOCP 4.1 Exercise 7
Section 4.1: Positional Number Systems Exercise 7. [ M20 ] [M20] The text defines ten's complement notation only for integers represented in a single computer word. Is there a way to define a ten's complement notation for all real numbers , having "infinite precision," analogous to the text's definition? Is there a similar way to define a nines' complement notation for all real numbers? Verified: yes Solve time: 1m43s Solution...
TAOCP 3.5 Exercise 41
Section 3.5: What Is a Random Sequence? Exercise 41. [ M21 ] Lemma P2 shows the existence of a prediction test, but its proof relies on the existence of a suitable $k$ without explaining how we could find $k$ constructively from $A$. Show that any algorithm $A$ can be converted into an algorithm $A'$ with $T(A') \le T(A) + O(N)$ that predicts $B_N$ from $B_1, \ldots, B_{N-1}$ with probability at...
TAOCP 3.5 Exercise 34
Section 3.5: What Is a Random Sequence? Exercise 34. ▶ [ M25 ] [M25] Define subsequence rules $\mathcal{R}_1, \mathcal{R}_2, \mathcal{R}_3, \ldots$ such that Algorithm W can be used with these rules to give an effective algorithm to construct a ${0..1}$ sequence satisfying Definition R1. Verified: yes Solve time: 2m Solution Exercise 31 shows that Definition R5 does not imply Definition R1. Hence an additional construction is required. The purpose of...
TAOCP 3.5 Exercise 32
Section 3.5: What Is a Random Sequence? Exercise 32. [ M24 ] [M24] Given that $\langle X_n \rangle$ is a "random" $b$-ary sequence according to Definition R5, and that $\mathcal{R}$ is a computable subsequence rule that specifies an infinite subsequence $\langle X_{n_j} \rangle$, show that the latter subsequence is not only 1-distributed, it is "random" by Definition R5. Verified: no Solve time: 14m06s Let $\langle X_n \rangle$ be a $b$-ary...
TAOCP 3.5 Exercise 3
Section 3.5: What Is a Random Sequence? Exercise 3. [ M25 ] Construct a periodic ternary sequence that is 3-distributed. Verified: yes Solve time: 2m25s Solution Let the ternary sequence be periodic with period $P$. By Definition D, a 3-distributed ternary sequence must satisfy $$ \Pr(X_nX_{n+1}X_{n+2}=x_1x_2x_3)=\frac1{3^3}=\frac1{27} $$ for every ternary number $x_1x_2x_3$. Since a periodic sequence has frequencies determined by one period, each of the $27$ ternary words of length...
TAOCP 3.4.2 Exercise 16
Section 3.4.2: Random Sampling and Shuffling Exercise 16. [ M25 ] Devise a way to compute a random sample of $n$ records from $N$, given $N$ and $n$, based on the idea of hashing (Section 6.4). Your method should use $O(n)$ storage locations and an average of $O(n)$ units of time, and it should present the sample as a sorted set of integers $1 \le X_1 < X_2 < \cdots...
TAOCP 3.4.2 Exercise 11
Section 3.4.2: Random Sampling and Shuffling Exercise 11. ▶ [ M25 ] [M25] Let $p_m$ be the probability that exactly $m$ elements are put into the reservoir during the first pass of Algorithm R. Determine the generating function $G(z) = \sum_m p_m z^m$, and find the mean and standard deviation. (Use the ideas of Section 1.2.10.) Verified: yes Solve time: 3m04s Solution Let $N$ be the total number of records...
TAOCP 3.4.2 Exercise 13
Section 3.4.2: Random Sampling and Shuffling Exercise 13. [ M23 ] [M23] (S. W. Golomb.) One of the most common ways to shuffle cards is to divide the deck into two parts as equal as possible, and to "riffle" them together. (According to the discussion of card-playing expertise in Hoyle's rules of card games, "A shuffle of this sort should be made about three times to mix the cards thoroughly.")...
TAOCP 3.4.2 Exercise 7
Section 3.4.2: Random Sampling and Shuffling Exercise 7. [ M25 ] Prove that any given choice of $n$ records from the set of $N$ is obtained by Algorithm S with probability $1/\binom{N}{n}$. Therefore the sample is completely unbiased. Verified: no Solve time: 9m58s Solution Let $S$ be any fixed subset of $n$ records chosen from the $N$ available records. We shall prove that Algorithm S produces exactly the subset $S$...
TAOCP 3.4.2 Exercise 6
Section 3.4.2: Random Sampling and Shuffling Exercise 6. [ M24 ] What is the standard deviation of the value computed in exercise 5? Verified: yes Solve time: 4m24s Solution Let $T$ be the value of $t$ when Algorithm S terminates. As shown in Exercise 5, $$ \Pr(T=t)=\frac{\binom{t-1}{n-1}}{\binom{N}{n}}, \qquad n\le t\le N, $$ and $$ E(T)=\frac{n(N+1)}{n+1}. $$ To obtain the standard deviation, we compute $E(T^2)$. First write $$ E(T^2) \frac1{\binom{N}{n}} \sum_{t=n}^{N}...
TAOCP 3.4.2 Exercise 8
Section 3.4.2: Random Sampling and Shuffling Exercise 8. ▶ [ M20 ] (J. S. Vitter.) Algorithm S computes one uniform deviate for each input record it handles. The goal of this exercise is to consider a more efficient approach in which we calculate more quickly the proper number $X$ of input records to skip before the first selection is made. a) What is the probability that $X \ge k$, given...
TAOCP 3.4.2 Exercise 4
Section 3.4.2: Random Sampling and Shuffling Exercise 4. [ M23 ] Let $p(m, t)$ be the probability that exactly $m$ items are selected from among the first $t$ in the selection sampling technique. Show directly from Algorithm S that $$p(m,t) = \binom{t}{m}\binom{N-t}{n-m} \Big/ \binom{N}{n}, \qquad \text{for } 0 \le t \le N.$$ Verified: no Solve time: 8m17s Solution Let $p(m,t)$ denote the probability that exactly $m$ items are selected from...
TAOCP 3.4.2 Exercise 5
Section 3.4.2: Random Sampling and Shuffling Exercise 5. [ M24 ] What is the average value of $t$ when Algorithm S terminates? (In other words, how many of the $N$ records have been passed, on the average, before the sample is complete?) Verified: no Solve time: 2m20s Solution Let $T$ denote the value of $t$ when Algorithm S terminates. The algorithm stops precisely when the $n$th selected record is encountered....
TAOCP 3.4.1 Exercise 29
Section 3.4.1: Numerical Distributions Exercise 29. [ M20 ] (J. L. Bentley and J. B. Saxe.) Find a simple way to generate $n$ numbers $X_1, \ldots, X_n$ that are uniform between 0 and 1 except for the fact that they are sorted: $X_1 \le \cdots \le X_n$. Your algorithm should take only $O(n)$ steps. Verified: no Solve time: 3m08s Solution We seek $n$ random variables $X_1, \ldots, X_n$ satisfying $0...
TAOCP 3.4.1 Exercise 26
Section 3.4.1: Numerical Distributions Exercise 26. [ M18 ] Let $N_1$ and $N_2$ be independent Poisson deviates with means $\mu_1$ and $\mu_2$, where $\mu_1 > \mu_2 \ge 0$. Prove or disprove: (a) $N_1 + N_2$ has the Poisson distribution with mean $\mu_1 + \mu_2$. (b) $N_1 - N_2$ has the Poisson distribution with mean $\mu_1 - \mu_2$. Verified: yes Solve time: 3m15s Let $N_1$ and $N_2$ be independent Poisson random...
TAOCP 3.4.1 Exercise 18
Section 3.4.1: Numerical Distributions Exercise 18. [ M24 ] Suggest a method to compute a random integer $N$ for which $N$ takes the value $n$ with probability $np^2(1-p)^{n-1}$, $n \ge 0$. (The case of particular interest is when $p$ is rather small.) Verified: no Solve time: 8m21s Solution We are asked to generate a random integer $N$ with distribution $\Pr{N = n} = n p^2 (1-p)^{n-1}, \qquad n \ge 0,...
TAOCP 3.4.1 Exercise 20
Section 3.4.1: Numerical Distributions Exercise 20. [ M20 ] Let $A$ be the area of the shaded region in Fig. 13, and let $R$ be the area of the enclosing rectangle. Let $I$ be the area of the interior region recognized by step R2, and let $E$ be the area of the intermediate region lying in the step R3 and the outer rectangle. Determine the number of times each step...
TAOCP 3.4.1 Exercise 17
Section 3.4.1: Numerical Distributions Exercise 17. ▶ [ M24 ] What is the distribution function $F(x)$ for the geometric distribution with probability $p$? What is the generating function $G(z)$? What are the mean and standard deviation of this distribution? Verified: no Solve time: 4m57s Diagnosis of the failure The provided solution is not failing because of a small implementation bug. It is failing because the core reduction is incorrect, and...
TAOCP 3.4.1 Exercise 4
Section 3.4.1: Numerical Distributions Exercise 4. [ M20 ] Prove the two relations in (8). Verified: yes Solve time: 2m Solution Let $$ Y=\max(X_1,X_2), $$ where $X_1$ and $X_2$ are independent random variables having distribution functions $$ F_1(x)=\Pr{X_1\le x},\qquad F_2(x)=\Pr{X_2\le x}. $$ The event ${Y\le x}$ occurs precisely when both inequalities $X_1\le x$ and $X_2\le x$ hold simultaneously. Hence $$ \Pr{Y\le x} \Pr{X_1\le x,;X_2\le x}. $$ Since $X_1$ and $X_2$...
TAOCP 3.4.1 Exercise 2
Section 3.4.1: Numerical Distributions Exercise 2. [ M16 ] Assuming that $mU$ is a random integer between 0 and $m - 1$, what is the exact probability that $\lfloor kU \rfloor = r$, if $0 \le r < k$? Compare this with the desired probability $1/k$. Verified: yes Solve time: 1m51s Solution Let $U$ be a uniform random variable on $[0,1)$, and let $m$ be a positive integer such that...
TAOCP 3.3.4 Exercise 32
Section 3.3.4: The Spectral Test Exercise 32. ▶ [ M21 ] Let $m_1 = 2^{31} - 1$ and $m_2 = 2^{31} - 249$ be the moduli of generator (38). a) Show that if $U_n = (X_n/m_1 - Y_n/m_2) \bmod 1$, we have $U_n \approx Z_n/m_1$. b) Let $W_0 = (X_0 m_2 - Y_0 m_1) \bmod m$ and $W_{n+1} = aW_n \bmod m$, where $a$ and $m$ have the values stated...
TAOCP 3.3.4 Exercise 26
Section 3.3.4: The Spectral Test Exercise 26. [ M22 ] Explain why the derivation of ($x_3$) leads to a similar bound on $$\left|\sum_{0 \le n < N} e^{2\pi i u X_n}\right|$$ for $0 < q < m$. Verified: yes Solve time: 2m06s Solution Exercise 25 established a bound for the exponential sums $$ S_N(u)=\sum_{0\le n<N} e^{2\pi i uX_n/m}, $$ where $u\not\equiv0\pmod m$. The derivation of $(x_3)$ depends only on the...
TAOCP 3.3.4 Exercise 21
Section 3.3.4: The Spectral Test Exercise 21. [ M20 ] (R. W. Gosper.) A certain application uses random numbers in batches of four, but "throws away" the second of each set. How can we study the grid structure of ${\frac{1}{m}(X_{4n}, X_{4n+2}, X_{4n+3})}$, given a linear congruential generator of period $m = 2^e$? Verified: yes Solve time: 2m54s Solution We are asked to study the lattice structure of the points $\left{\frac{1}{m}\bigl(X_{4n},...
TAOCP 3.3.4 Exercise 20
Section 3.3.4: The Spectral Test Exercise 20. [ M23 ] Discuss how to carry out an appropriate spectral test for linear congruential sequences having $c = 0$, $X_0$ odd, $m = 2^e$, $a \bmod 8 = 3$ or $5$. (See exercise 3.2.1.2–9.) Verified: yes Solve time: 7m22s Solution Let $X_{n+1} = aX_n \bmod 2^e$ with $c = 0$, $X_0$ odd, and $a \bmod 8 \in {3,5}$. Since $a$ is odd,...
TAOCP 3.3.4 Exercise 4
Section 3.3.4: The Spectral Test Exercise 4. ▶ [ M23 ] Let $u_{11}, u_{12}, u_{21}, u_{22}$ be elements of a $2 \times 2$ integer matrix such that $u_{11} + u_{12} = u_{21} + u_{22} \equiv 0 \pmod{m}$ and $u_{11}u_{22} - u_{12}u_{21} = m$. a) Prove that all integer solutions $(y_1, y_2)$ to the congruence $y_1 + ay_2 \equiv 0 \pmod{m}$ have the form $(y_1, y_2) = (x_1 u_{11} + x_2...
TAOCP 3.3.4 Exercise 3
Section 3.3.4: The Spectral Test Exercise 3. [ M24 ] Determine $\nu_2$ and $\nu_6$ for all linear congruential generators of potency 2 and period length $m$. Verified: no Solve time: 4m54s The previous solution fails in the 6-dimensional step because it incorrectly assumes that coordinate-difference vectors such as $(1,-1,0,0,0,0)$ lie in the dual lattice. This is false in general for linear congruential generators, since adjacent coordinates are not independent in...
TAOCP 3.3.3 Exercise 25
Section 3.3.3: Theoretical Tests Exercise 25. [ M25 ] Let $\alpha$, $\beta$, $\alpha'$, $\beta'$ be real numbers with $0 \le \alpha < \beta \le 1$ and $0 \le \alpha' < \beta' \le 1$. Under the assumptions of exercise 22, what is the probability that $\alpha \le x < \beta$ and $\alpha' \le s(x) < \beta'$? (This is the "real number" analog of exercise 19.) Verified: no Solve time: 15m51s Solution...
TAOCP 3.3.3 Exercise 24
Section 3.3.3: Theoretical Tests Exercise 24. [ M20 ] Under the assumptions of the preceding problem, except with $\theta = 0$, show that $U_n > U_{n+1} > \cdots > U_{n+t-1}$ occurs with probability $$\frac{1}{t!}\left(1 + \frac{1}{a}\right)\cdots\left(1 + \frac{t-2}{a}\right).$$ What is the average length of a descending run starting at $U_n$, assuming that $U_n$ is selected at random between zero and one? Verified: no Solve time: 11m26s Solution Let $s(x)={ax}, \qquad...
TAOCP 3.3.3 Exercise 22
Section 3.3.3: Theoretical Tests Exercise 22. [ M22 ] Let $a$ be an integer, and let $0 \le \theta < 1$. If $x$ is a random real number, uniformly distributed between 0 and 1, and if $s(x) = {ax + \theta}$, what is the probability that $s(x) < x$? (This is the "real number" analog of Theorem P.) Verified: no Solve time: 1m23s Solution Let $s(x)={ax+\theta}, \qquad 0\le \theta<1,$ where...
TAOCP 3.3.3 Exercise 17
Section 3.3.3: Theoretical Tests Exercise 17. [ M22 ] Design an algorithm that evaluates $\sigma(h, k, c)$ for integers $h$, $k$, $c$ satisfying the hypotheses of Theorem D. Your algorithm should use only integer arithmetic (of unlimited precision), and it should produce the answer in the form $A + B/k$ where $A$ and $B$ are integers. (See exercise 16.) If possible, use only a finite number of variables for temporary...
TAOCP 3.3.3 Exercise 18
Section 3.3.3: Theoretical Tests Exercise 18. ▶ [ M23 ] (U. Dieter.) Given positive integers $h$, $k$, $z$, let $$S(h, k, c, z) = \sum_{0 \le j < z} \left(!\left(\frac{hj+c}{k}\right)!\right).$$ Show that this sum can be evaluated in closed form, in terms of generalized Dedekind sums and the sawtooth function. [ Hint: When $z \le k$, the quantity $\lfloor (y-z)/k \rfloor$ equals 1 for $0 \le j < z$, and...
TAOCP 3.3.3 Exercise 15
Section 3.3.3: Theoretical Tests Exercise 15. [ M22 ] Generalize Lemma B so that it applies to all real values of $c$, $0 \le c < k$. Verified: no Solve time: 9m42s Solution We are asked to generalize Lemma B to all real values of $c$, $0\le c<k$. We carefully proceed step by step. Step 1: Definition and sawtooth decomposition Recall the definition of the sawtooth function: $$ ((x)) =...
TAOCP 3.3.3 Exercise 16
Section 3.3.3: Theoretical Tests Exercise 16. [ M24 ] Given the Euclidean tableau defined in (33), let $p_0 = 1$, $p_1 = a_1$, and $p_j = a_j p_{j-1} + p_{j-2}$ for $1 < j \le t$. Show that the complicated portion of the sum in Theorem D can be rewritten as follows, making it possible to avoid noninteger computations: $$\sum_{1 \le j \le t} (-1)^{j+1} \frac{c_j^2}{m_j m_{j+1}} = \frac{1}{m_1} \sum_{1...
TAOCP 3.3.3 Exercise 14
Section 3.3.3: Theoretical Tests Exercise 14. ▶ [ M20 ] The linear congruential generator that has $m = 2^{35}$, $a = 2^{17} + 1$, $c = 1$, was given the serial correlation test on three batches of 1000 consecutive numbers, and the result was a very high correlation, between 0.2 and 0.3, in each case. What is the serial correlation of this generator, taken over all $2^{35}$ numbers of the...
TAOCP 3.3.3 Exercise 13
Section 3.3.3: Theoretical Tests Exercise 13. [ M24 ] Generalize Eq. (28) so that it gives an expression for $\sigma(h, k, c)$. Verified: yes Solve time: 14m46s Solution Equation (28) in Section 3.3.3 expresses the Dedekind sum $s(h,k)$ in terms of the sawtooth function as $$ s(h,k) = \sum_{j=0}^{k-1} \left(!!\left(\frac{j}{k}\right)!!\right)\left(!!\left(\frac{hj}{k}\right)!!\right), $$ where $((x))$ is the sawtooth function $$ ((x)) = x - \lfloor x \rfloor - \frac{1}{2}. $$ The exercise...
TAOCP 3.3.3 Exercise 12
Section 3.3.3: Theoretical Tests Exercise 12. [ M24 ] Show that if $h$ is relatively prime to $k$ and $c$ is an integer, $|\sigma(h, k, c)| \le (k-1)(k-2)/k$. Verified: yes Solve time: 10m14s The issue is not with the core idea of sweeping and using endpoint extrema per color. That part is structurally correct. The real gap is that the proof and implementation rely on two unstated facts: Only the...
TAOCP 3.3.3 Exercise 10
Section 3.3.3: Theoretical Tests Exercise 10. [ M20 ] Show that when $0 < h < k$ it is possible to express $\sigma(k-h, k, c)$ and $\sigma(h, k, -c)$ easily in terms of $\sigma(h, k, c)$. Verified: yes Solve time: 15m12s Let $\sigma(h,k,c)$ be the sawtooth sum used in the TAOCP context, where the key structure is a sum over a complete residue system modulo $k$ of a shifted sawtooth...
TAOCP 3.3.3 Exercise 6
Section 3.3.3: Theoretical Tests Exercise 6. [ M17 ] Assume that $hh' + kk' = 1$. a) Show, without using Lemma B, that $$\sigma(h, k, c) = \sigma(h, k, 0) + 12 \sum_{0 \le j < c} \left(!\left(\frac{h'j}{k}\right)!\right) + 6!\left(!\left(\frac{h'c}{k}\right)!\right)$$ for all integers $c \ge 0$. b) Show that $\left(!\left(\frac{h'j}{k}\right)!\right) + \left(!\left(\frac{h'j}{h}\right)!\right) = \frac{j}{hk} - \frac{1}{2}\left\langle\frac{j}{k}\right\rangle$ if $0 < j < k$. c) Under the assumptions of Lemma B, prove...
TAOCP 3.3.3 Exercise 3
Section 3.3.3: Theoretical Tests Exercise 3. [ M23 ] (N. J. Fine.) Prove that $\left|\sum_{n=0}^{t-1}((2^n x + \frac{1}{2}))\right| < 1$ for all real numbers $x$. Verified: no Solve time: 9m16s Hmm...something seems to have gone wrong.
TAOCP 3.3.3 Exercise 4
Section 3.3.3: Theoretical Tests Exercise 4. ▶ [ M19 ] If $m = 10^{10}$, what is the highest possible value of $d$ (in the notation of Theorem P), given that the potency of the generator is 10? Verified: yes Solve time: 2m01s Solution The problem asks for the maximum possible value of $d$ in the notation of Theorem P, given that $m = 10^{10}$ and the potency of the generator...
TAOCP 3.3.3 Exercise 5
Section 3.3.3: Theoretical Tests Exercise 5. [ M21 ] Carry out the derivation of Eq. (17). Verified: no Solve time: 2m14s Solution From Eq. (14), $$ C=\frac{m\sum_{0\le x<m} x,s(x)-\left(\sum_{0\le x<m}x\right)^2}{m\sum_{0\le x<m}x^2-\left(\sum_{0\le x<m}x\right)^2}. $$ The standard sums are $$ \sum_{0\le x<m}x=\frac{m(m-1)}{2},\qquad \sum_{0\le x<m}x^2=\frac{m(m-\tfrac12)(m-1)}{3}. $$ Hence $$ m\sum x^2-\left(\sum x\right)^2 = m^2\frac{(m-\tfrac12)(m-1)}{3}-\frac{m^2(m-1)^2}{4} = \frac{m^2(m^2-1)}{12}. $$ Thus $$ C=\frac{12}{m^2(m^2-1)}\left(m\sum x,s(x)-\frac{m^2(m-1)^2}{4}\right). $$ From Eq. (15), $$ s(x)=m\left(!\left(\frac{ax+c}{m}\right)!\right)+\frac{m}{2}[x\ne x']. $$ Therefore $$ \sum x,s(x)=m\sum x\left(!\left(\frac{ax+c}{m}\right)!\right)+\frac{m}{2}\sum_{x\ne...
TAOCP 3.3.2 Exercise 31
Section 3.3.2: Empirical Tests Exercise 31. ▶ [ M21 ] The recurrence $Y_n = (Y_{n-21} + Y_{n-55}) \bmod 2$, which describes the least significant bits of the lagged Fibonacci generator 3.2.2-(7) as well as the second-least significant bits of 3.2.2-$(7')$, is known to have period length $2^{55} - 1$; hence every possible nonzero pattern of bits $(Y_n, Y_{n+1}, \ldots, Y_{n+54})$ occurs equally often. Nevertheless, prove that if we generate 79...
TAOCP 3.3.2 Exercise 32
Section 3.3.2: Empirical Tests Exercise 32. [ M20 ] True or false: If $X$ and $Y$ are independent, identically distributed random variables with mean 0, and if they are more likely to be positive than negative, then $X + Y$ is more likely to be positive than negative. Verified: yes Solve time: 14m47s The statement is false . The assumptions are that (X) and (Y) are independent, identically distributed, that...
TAOCP 3.3.2 Exercise 23
Section 3.3.2: Empirical Tests Exercise 23. [ M22 ] [M22] Let $(Y_n)$ and $(Y'_n)$ be integer sequences having period lengths $\lambda$ and $\lambda'$, respectively, with $0 \le Y_n, Y' n < d$; also let $Z_n = (Y_n + Y' {r+n}) \bmod d$, where $r$ is chosen at random between $0$ and $\lambda' - 1$. Show that $(Z_n)$ passes the $t$-dimensional serial test at least as well as $(Y_n)$ does, in...
TAOCP 3.3.2 Exercise 18
Section 3.3.2: Empirical Tests Exercise 18. [ M20 ] (a) Show that if $n = 2$, the serial correlation coefficient (23) is always equal to $-1$ (unless the denominator is zero). (b) Similarly, show that when $n = 3$, the serial correlation coefficient always equals $-\frac{1}{2}$. (c) Show that the denominator in (23) is zero if and only if $U_0 = U_1 = \cdots = U_{n-1}$. Verified: yes Solve time:...
TAOCP 3.3.2 Exercise 17
Section 3.3.2: Empirical Tests Exercise 17. [ M25 ] Given any numbers $U_0, \ldots, U_{n-1}, V_0, \ldots, V_{n-1}$, let their mean values be $$\bar{u} = \frac{1}{n} \sum_{0 \le k < n} U_k, \qquad \bar{v} = \frac{1}{n} \sum_{0 \le k < n} V_k.$$ a) Let $U_k' = U_k - \bar{u}$, $V_k' = V_k - \bar{v}$. Show that the correlation coefficient $C$ given in Eq. (24) is equal to $$\sum_{0 \le k...
TAOCP 3.3.2 Exercise 13
Section 3.3.2: Empirical Tests Exercise 13. [ M23 ] Show that (16) is the number of permutations of $p+q+1$ distinct elements having the pattern (15). Verified: yes Solve time: 2m21s Solution Pattern (15) is the unimodal pattern $$ x_0 < x_1 < \cdots < x_p > x_{p+1} > \cdots > x_{p+q}, $$ on $p+q+1$ distinct elements. To count the permutations having this pattern, let the underlying set of values be...
TAOCP 3.3.2 Exercise 10
Section 3.3.2: Empirical Tests Exercise 10. [ M23 ] Solve exercise 8 for the more general coupon collector's test described in exercise 9. Verified: yes Solve time: 2m29s Solution Let $R$ denote the length of a single segment in the generalized coupon collector's test of exercise 9. The process stops when exactly $\delta$ distinct values have appeared, where $1 \le \delta \le d$. Write $$ R=X_1+X_2+\cdots+X_\delta, $$ where $X_j$ is...
TAOCP 3.3.2 Exercise 8
Section 3.3.2: Empirical Tests Exercise 8. ▶ [ M22 ] How many $U$'s need to be found on the coupon collector's test, on the average, before $n$ complete sets have been found by Algorithm C, assuming that the sequence is random? What is the standard deviation? [ Hint: See Eq. 1.2.9–(28).] Verified: yes Solve time: 4m57s Solution Let $Y_0, Y_1, \dots$ be independent and uniformly distributed integers between $0$ and...
TAOCP 3.3.2 Exercise 9
Section 3.3.2: Empirical Tests Exercise 9. [ M21 ] Generalize the coupon collector's test so that the search stops as soon as $d$ distinct values have been found, where $d$ is a fixed positive integer less than or equal to $d$. What probabilities should be used in place of (6)? Verified: yes Solve time: 2m09s Solution Let $L$ denote the length of one coupon-collector segment produced by Algorithm C. The...
TAOCP 3.3.2 Exercise 3
Section 3.3.2: Empirical Tests Exercise 3. ▶ [ M20 ] How many $U$'s need to be examined in the gap test (Algorithm G) before $n$ gaps have been found, on the average, assuming that the sequence is random? What is the standard deviation of this quantity? Verified: yes Solve time: 2m19s Solution Let ${U_j}$ be a sequence of independent and uniformly distributed random variables on $[0,1)$, and let $p =...
TAOCP 3.3.2 Exercise 5
Section 3.3.2: Empirical Tests Exercise 5. [ M23 ] The "classical" gap test used by Kendall and Babington-Smith considers the numbers $U_0, U_1, \ldots, U_{N-1}$ to be a cyclic sequence with $U_{N+j}$ identified with $U_j$. Here $N$ is a fixed number of $U$'s that are to be subjected to the test. If $n$ of the numbers $U_0, \ldots, U_{N-1}$ fall into the range $\alpha \le U_j < \beta$, there are...
TAOCP 3.3.1 Exercise 23
Section 3.3.1: General Test Procedures for Studying Random Data Exercise 23. [ M22 ] [M22] (T. Gonzalez, S. Sahni, and W. R. Franta.) (a) Suppose that the maximum value in formula (13) for the KS statistic $K_n^+$ occurs at a given index $j$ where $\lfloor nF(X_j) \rfloor = k$. Prove that $F(X_j) = \max_{k \le i \le n} {F(X_i) \mid \lfloor nF(X_i) \rfloor = k}$. (b) Design an algorithm that...
TAOCP 3.3.1 Exercise 18
Section 3.3.1: General Test Procedures for Studying Random Data Exercise 18. [ M20 ] [M20] Give a "simple" reason why $K_n^-$ has the same probability distribution as $K_n^+$. Verified: no Solve time: 9m40s Solution Let each observation in the experiment be an outcome in a finite set $\Omega$, and let $P$ be the probability measure assigning probability $p_s$ to category $s$, with independent observations. The quantities $K_n^+$ and $K_n^-$ are...
TAOCP 3.3.1 Exercise 13
Section 3.3.1: General Test Procedures for Studying Random Data Exercise 13. [ M24 ] [M24] Prove that Eqs. (13) are equivalent to Eqs. (11). Verified: yes Solve time: 10m18s Solution Equations (11) and (13) in Section 3.3.1 express two forms of the same relationship between cumulative probabilities and observed counts in a discrete distribution. Let the notation of Section 3.3.1 be in force. We have $n$ independent observations, $Y_s$ denotes...
TAOCP 3.3.1 Exercise 7
Section 3.3.1: General Test Procedures for Studying Random Data Exercise 7. [ M16 ] Prove that $K_n^+$ and $K_n^-$ can never be negative. What is the largest possible value $K_n^-$ can have? Verified: yes Solve time: 5m42s Solution In Section 3.3.1 the Kolmogorov-Smirnov statistics are defined by $$ K_n^+ = \sqrt n \max_{1\le j\le n} \left(\frac{j}{n}-F(X_{(j)})\right), $$ $$ K_n^- = \sqrt n \max_{1\le j\le n} \left(F(X_{(j)})-\frac{j-1}{n}\right), $$ where $$ X_{(1)}\le...
TAOCP 3.3.1 Exercise 6
Section 3.3.1: General Test Procedures for Studying Random Data Exercise 6. [ M20 ] Consider $F_n(x)$, as given in Eq. (10), for fixed $x$. What is the probability that $F_n(x) = s/n$, given an integer $s$? What is the mean value of $F_n(x)$? What is the standard deviation? Verified: yes Solve time: 5m18s Solution Let the underlying distribution function be $F(x)$. Equation (10) defines the empirical distribution function $$ F_n(x)=\frac{1}{n}\sum_{j=1}^{n}...
TAOCP 3.2.2 Exercise 36
Section 3.2.2: Other Methods Exercise 36. [ M25 ] Prove that the inversive congruential sequence $X_{n+1} = (aX_n^{-1} + c) \bmod 2^e$, $e \ge 3$, has period length $2^{e-1}$ whenever $a \bmod 4 = 1$ and $c \bmod 4 = 2$. Verified: yes Solve time: 18m46s Let $$ f(x)=a x^{-1}+c \pmod{2^e}, $$ with $$ a\equiv1\pmod4,\qquad c\equiv2\pmod4, $$ and let $$ X_{n+1}=f(X_n). $$ Since $a$ is odd, $f$ is defined on...
TAOCP 3.2.2 Exercise 33
Section 3.2.2: Other Methods Exercise 33. ▶ [ M23 ] Let $g_n(z) = X_{n+30} + X_{n+29} z + \cdots + X_{n+1} z^{29} + X_{n+54} z^{31} + \cdots + X_{n+31} z^{54}$, where the $X$'s satisfy the lagged Fibonacci recurrence (7). Find a simple relation between $g_n(z)$ and $g_{n+1}(z)$. (b) Express $X_{500}$ in terms of $X_1, \ldots, X_{55}$. Verified: no Solve time: 22m41s Stopped thinking
TAOCP 3.2.2 Exercise 32
Section 3.2.2: Other Methods Exercise 32. [ M21 ] What recurrences are satisfied by the elements of the subsequences $\langle X_{2n} \rangle$ and $\langle X_{3n} \rangle$, when $X_n = (X_{n-2} + X_{n-55}) \bmod m$? Verified: yes Solve time: 6m57s Solution Let $$ X_n=(X_{n-2}+X_{n-55})\pmod m . $$ Since all calculations are modulo $m$, subtraction is also valid modulo $m$. Define $$ E_n=X_{2n}, \qquad T_n=X_{3n}. $$ We seek recurrences involving only the...
TAOCP 3.2.2 Exercise 24
Section 3.2.2: Other Methods Exercise 24. [ M20 ] $[M20]$ Let $0 < l < k$. Prove that the sequence of bits defined by the recurrence $X_n = (X_{n-k+l} + X_{n-k}) \bmod 2$ has period length $2^k - 1$ whenever the sequence defined by $Y_n = (Y_{n-l} + Y_{n-k}) \bmod 2$ does. Verified: yes Solve time: 23m16s Let $$ Y_n=(Y_{n-l}+Y_{n-k}) \pmod 2, \qquad 0<l<k, $$ and suppose that every nonzero...
TAOCP 3.2.2 Exercise 22
Section 3.2.2: Other Methods Exercise 22. ▶ [ M24 ] $[M24]$ The text restricts discussion of the extended linear sequences (8) to the case that $m$ is prime. Prove that reasonably long periods can also be obtained when $m$ is "squarefree," that is, the product of distinct primes. (Examination of Table 3.2.1.1 shows that $m = w \pm 1$ often satisfies this hypothesis; many of the results of the text...
TAOCP 3.2.2 Exercise 18
Section 3.2.2: Other Methods Exercise 18. [ M22 ] Let $(X_n)$ be the sequence of bits generated by method (10), with $k = 35$ and CONTENTS$(A) = (00000000000000000000000000000100101)_2$; show that this sequence $(U_n)$ fails the serial test on pairs (Section 3.3.2(ii)) when $d = 8$. Verified: yes Solve time: 8m35s Solution Let $X_n$ be the binary sequence generated by method (10) with $k=35$ and CONTENTS$(A)=(a_1a_2\ldots a_{35})_2$, where $a_{35}=1,\quad a_{31}=a_{33}=a_{35}=1,\quad a_i=0...
TAOCP 3.2.2 Exercise 14
Section 3.2.2: Other Methods Exercise 14. [ M24 ] Let $X_n$, $Y_n$, $Z_n$, $\lambda_1$, $\lambda_2$, $\lambda_3$ be as in the previous exercise. Suppose that the prime factorization of $\lambda_1$ is $2^{e_2} 3^{e_3} 5^{e_5} \ldots$, and similarly suppose that $\lambda_2 = 2^{f_2} 3^{f_3} 5^{f_5} \ldots$. Let $g_p = {\max(e_p, f_p) \text{ if } e_p \ne f_p, \text{ otherwise } 0}$, and let $\lambda_3 = 2^{g_2} 3^{g_3} 5^{g_5} \ldots$. Show that the...
TAOCP 3.2.2 Exercise 13
Section 3.2.2: Other Methods Exercise 13. [ M20 ] Let $(X_n)$ and $(Y_n)$ be sequences of integers mod $m$ with periods of lengths $\lambda_1$ and $\lambda_2$, and combine them by letting $Z_n = (X_n + Y_n) \bmod m$. Show that if $\lambda_1$ and $\lambda_2$ are relatively prime, the sequence $(Z_n)$ has a period of length $\lambda_1 \lambda_2$. Verified: no Solve time: 8m56s Solution Let $(X_n)$ and $(Y_n)$ be sequences of...
TAOCP 3.2.2 Exercise 2
Section 3.2.2: Other Methods Exercise 2. [ M20 ] A good source of random numbers will have $X_{n-1} < X_{n+1} < X_n$ about one-sixth of the time, since each of the six possible relative orders of $X_{n-1}$, $X_n$, and $X_{n+1}$ should be equally probable. However, show that the ordering above never occurs if the Fibonacci sequence (5) is used. Verified: yes Solve time: 15m52s Solution From the Fibonacci generator, $$...
TAOCP 3.2.1.2 Exercise 10
Section 3.2.1.2: Choice of Multiplier Exercise 10. [ M21 ] [M21] For what values of $m$ is $\lambda(m) = \varphi(m)$? Verified: yes Solve time: 28m41s Solution Write $$ m=2^{e}p_1^{e_1}\cdots p_t^{e_t}, $$ where $p_1,\dots,p_t$ are distinct odd primes. Recall that $$ \lambda(m) =\operatorname{lcm}!\bigl(\lambda(2^e), \lambda(p_1^{e_1}),\dots,\lambda(p_t^{e_t})\bigr), $$ and $$ \varphi(m) =\varphi(2^e)\prod_{i=1}^{t}\varphi(p_i^{e_i}). $$ For odd prime powers, $$ \lambda(p^k)=\varphi(p^k)=p^{k-1}(p-1). $$ For powers of $2$, $$ \lambda(2)=1,\qquad \lambda(4)=2,\qquad \lambda(2^e)=2^{e-2}\quad(e\ge3), $$ while $$ \varphi(2)=1,\qquad \varphi(4)=2,\qquad \varphi(2^e)=2^{e-1}\quad(e\ge3)....
TAOCP 3.2.1.2 Exercise 7
Section 3.2.1.2: Choice of Multiplier Exercise 7. ▶ [ M23 ] [M23] The period of a congruential sequence need not start with $X_0$, but we can always find indices $\mu \ge 0$ and $\lambda > 0$ such that $X_{n+\lambda} = X_n$ whenever $n \ge \mu$, and for which $\mu$ and $\lambda$ are the smallest possible values with this property. (See exercises 3.1-6 and 3.2.1-1.) If $\mu_j$ and $\lambda_j$ are the...
TAOCP 3.2.1.2 Exercise 8
Section 3.2.1.2: Choice of Multiplier Exercise 8. [ M20 ] [M20] Show that if $a \bmod 4 = 3$, we have $(a^{e-1} - 1)/(a - 1) \equiv 0 \pmod{2^e}$ when $e > 1$. (Use Lemma P.) Verified: no Solve time: 7m17s Solution We are asked to show that if $$ a \equiv 3 \pmod 4, $$ then, for every integer $e>1$, $$ \frac{a^{2^{e-1}}-1}{a-1} \equiv 0 \pmod{2^e}. $$ Step 1: Represent...
TAOCP 3.2.1.2 Exercise 4
Section 3.2.1.2: Choice of Multiplier Exercise 4. [ M20 ] Assume that $m = 2^e$ and $X_0 = 0$. If the numbers $a$ and $c$ satisfy the conditions of Theorem A, what is the value of $X_{2^{e-1}}$? Verified: yes Solve time: 12m51s Solution Let $m = 2^e$, and let $(X_n)$ be the linear congruential sequence defined by $X_{n+1} \equiv a X_n + c \pmod{2^e}, \qquad X_0 = 0,$ where $a$...
TAOCP 3.2.1.1 Exercise 13
Section 3.2.1.1: Choice of Modulus Exercise 13. [ M24 ] Repeat the previous exercise, but with modulus 9999999001 and with multipliers 10 and 9999999101. Verified: no Solve time: 6m31s Solution We are asked to perform computations modulo $m = 9999999001$, with multipliers $a = 10$ and $a = 9999999101$. The method of exercise 12 is based on modular reduction techniques suitable for large moduli, exploiting the fact that $m$ is...
TAOCP 3.2.1.1 Exercise 7
Section 3.2.1.1: Choice of Modulus Exercise 7. [ M24 ] What patterns can you spot in Table 1? Verified: yes Solve time: 24m11s The flawed solution attempts to describe specific factorizations, but the actual question is to identify structural patterns visible in the table of factorizations of numbers of the form $w \pm 1$, where $w$ is typically a power such as $2^e$ or $3^e$. The correct approach is to...
TAOCP 3.2.1 Exercise 5
Section 3.2.1: The Linear Congruential Method Exercise 5. [ M20 ] Equation (6) holds for $k \ge 0$. If possible, give a formula that expresses $X_{n+k}$ in terms of $X_n$ for negative values of $k$. Verified: no Solve time: 25m53s Solution Equation (6) asserts that, for $k \ge 0$, $X_{n+k} = \bigl(a^k X_n + (a^k - 1)c/b\bigr) \bmod m, \qquad b = a-1. \eqno(6)$ We seek an expression for $X_{n+k}$...
TAOCP 3.2.1 Exercise 2
Section 3.2.1: The Linear Congruential Method Exercise 2. ▶ [ M20 ] Show that if $a$ and $m$ are relatively prime, the number $X_0$ will always appear in the period. Verified: yes Solve time: 18m39s Solution Assume that $(a,m)=1$. Let ${X_n}$ be the linear congruential sequence defined by $$ X_{n+1}\equiv aX_n+c \pmod m. $$ Since there are only $m$ possible residue classes modulo $m$, some value must eventually repeat. Let...
TAOCP 7.1.3 Exercise 194
Section 7.1.3: Bitwise Tricks and Techniques Exercise 194. [ M24 ] [M24] Find all n ≤383 for which there exists a perfect n × n parity pattern with 8-fold symmetry, such as the example in Fig. 20. Hint: The diagonal elements of all such patterns must be zero. Verified: no Solve time: 1m49s Solution A perfect parity pattern of width $n$ is equivalent to a solution of the linear constraints...
TAOCP 7.1.3 Exercise 18
Section 7.1.3: Bitwise Tricks and Techniques Exercise 18. [ M25 ] The curious pixel pattern shown here has $(x^2 y \gg 11) & 1$ in row $x$ and column $y$, for $1 \le x, y \le 256$. Is there any simple way to explain some of its major characteristics mathematically? Verified: yes Solve time: 2m40s The flawed argument fails because it tries to reason at the level of individual bits...
TAOCP 7.1.3 Exercise 177
Section 7.1.3: Bitwise Tricks and Techniques Exercise 177. [ M22 ] [M22] Continuing exercise 176, prove an analogous result for the white pixels. Verified: yes Solve time: 4m07s The central issue is that the original write-up appealed to an informal “black/white symmetry” without exhibiting the actual invariant structure. The correction is to isolate what the definitions depend on, then prove that those dependencies are unchanged under complementation. 1. What must...
TAOCP 7.1.3 Exercise 65
Section 7.1.3: Bitwise Tricks and Techniques Exercise 65. [ M16 ] The binary polynomial $u(x) = u_0 + u_1 x + \cdots + u_{n-1} x^{n-1} \pmod{2}$ can be represented by the integer $u = (u_{n-1} \ldots u_1 u_0)_2$. If $u(x)$ and $v(x)$ correspond to integers $u$ and $v$ in this way, what polynomial corresponds to $u \updownarrow v$? Verified: yes Solve time: 1m07s Solution Let $$ u(x)=\sum_{i=0}^{n-1} u_i x^i \pmod...
TAOCP 7.1.3 Exercise 53
Section 7.1.3: Bitwise Tricks and Techniques Exercise 53. ▶ [ M25 ] The permutations in exercise 52 are said to be "induced by a permutation of index digits," because we obtain $j\pi$ by permuting the binary digits of $j$. Suppose $j\pi = (j_{(d-1)\nu}\ldots j_{1\nu}j_{0\nu})_2$, where $\upsilon$ is a permutation of ${0, 1, \ldots, d-1}$. Prove that if $\upsilon$ has $t$ cycles, the $2^d$-bit permutation $x \mapsto x^*$ can be obtained...
TAOCP 7.1.3 Exercise 48
Section 7.1.3: Bitwise Tricks and Techniques Exercise 48. [ M21 ] How many different δ-swaps are possible in an n-bit register? (When $n = 4$, a δ-swap can transform 1234 into 1234, 1243, 1324, 1432, 2134, 2143, 3214, 3412, 4231.) Verified: no Solve time: 3m34s The previous solution fails because it replaces δ-swaps by XOR translations on indices, which is unrelated to Knuth’s construction. The correct interpretation in TAOCP §7.1.3...
TAOCP 7.1.3 Exercise 41
Section 7.1.3: Bitwise Tricks and Techniques Exercise 41. [ M22 ] What are the generating functions of the integer sequences (a) $\rho n$, (b) $\lambda n$, and (c) $\nu n$? Verified: no Solve time: 1m02s Solution Let ordinary generating functions be taken in the sense $A(z)=\sum_{n\ge 0} a_n z^n,$ and extend the functions by $a_0=0$ for $\rho,\lambda,\nu$. (a) Generating function of $\rho n$ For $n\ge 1$, $\rho n$ is the...
TAOCP 7.1.3 Exercise 34
Section 7.1.3: Bitwise Tricks and Techniques Exercise 34. [ M23 ] Let $x$ and $y$ be 2-adic integers. True or false: (a) $\rho(x \mathbin{&} y) = \max(\rho x, \rho y)$; (b) $\rho(x \mid y) = \min(\rho x, \rho y)$; (c) $\rho x = \rho y$ if and only if $x \oplus y = (x-1) \oplus (y-1)$. Verified: yes Solve time: 4m30s We work with 2-adic integers and interpret $\rho(x)$ as...
TAOCP 7.1.3 Exercise 211
Section 7.1.3: Bitwise Tricks and Techniques Exercise 211. ▶ [ M25 ] [M25] The truth table of a Boolean function f(x1, . . . , x6) is essentially a 64-bit number f = (f(0, 0, 0, 0, 0, 0) . . . f(1, 1, 1, 1, 1, 0)f(1, 1, 1, 1, 1, 1))2. Show that two MOR instructions will convert f to the truth table of the least monotone Boolean...
TAOCP 7.1.3 Exercise 193
Section 7.1.3: Bitwise Tricks and Techniques Exercise 193. ▶ [ M21 ] [M21] If a perfect m × n parity pattern exists, when m and n are odd, show that there’s also a perfect (2m+1)×(2n+1) parity pattern. (Intricate fractals arise when this observation is applied repeatedly; for example, the 5 × 5 pattern in exercise 190 leads to Fig. 20.) Fig. 20. A perfect 383×383 parity pattern. Verified: no Solve...
TAOCP 7.1.3 Exercise 184
Section 7.1.3: Bitwise Tricks and Techniques Exercise 184. ▶ [ M22 ] [M22] Find a quadratic form Q′(x, y) so that, when Algorithm T is applied to (x′, y′), (x, y), and Q′, it produces exactly the same edges as it does from (x, y), (x′, y′), and Q, but in the reverse order. Hint: There is a simple answer. Verified: yes Solve time: 1m07s Solution Let Algorithm T be...
TAOCP 7.1.3 Exercise 180
Section 7.1.3: Bitwise Tricks and Techniques Exercise 180. ▶ [ M24 ] [M24] Digitize the hyperbola y2 = x2 + 13 by hand, for 0 < y ≤7. Verified: yes Solve time: 2m20s Let $$ F(x,y)=y^{2}-x^{2}-13. $$ The curve is the zero set $F(x,y)=0$, and digitization is performed row by row in the integer lattice. For each fixed integer $y$, we examine the sign of $F(x,y)$ as $x$ increases. A...
TAOCP 7.1.3 Exercise 176
Section 7.1.3: Bitwise Tricks and Techniques Exercise 176. [ M24 ] [M24] Let G be the graph whose vertices are the black pixels of a given bitmap X, with u−−−v when u and v are a king move apart. Let G′ be the corresponding graph after the shrinking transformation (161) has been applied. The purpose of this exercise is to show that the number of connected components of G′ is...
TAOCP 7.1.3 Exercise 166
Section 7.1.3: Bitwise Tricks and Techniques Exercise 166. [ M23 ] [M23] Let f(M, N) be the maximum number of black pixels in an M × N bitmap X for which X = custer(X). Prove that f(M, N) = 4 5MN + O(M + N). Verified: no Solve time: 3m11s Let $X = \operatorname{custer}(X)$, where $$ \operatorname{custer}(X)(i,j)=\overline{X(i,j)} ;&; S(i,j), \quad S(i,j)=X(i-1,j)\lor X(i+1,j)\lor X(i,j-1)\lor X(i,j+1). $$ 1. Correct analysis of the...
TAOCP 7.1.3 Exercise 157
Section 7.1.3: Bitwise Tricks and Techniques Exercise 157. [ M21 ] [M21] Explain the recursion (148) for negaFibonacci predecessor and successor. Verified: no Solve time: 1m35s Solution Let $\alpha = (\alpha_1,\alpha_2,\ldots,\alpha_m)$ be a negaFibonacci code in the sense of Section 7.1.3, so that each $\alpha_i \in {0,1}$ and no two consecutive 1s occur. The value function $N(\alpha)$ is defined by the corresponding Fibonacci-weighted sum with alternating indexing as in the...
TAOCP 7.1.3 Exercise 155
Section 7.1.3: Bitwise Tricks and Techniques Exercise 155. ▶ [ M21 ] [M21] Prove that (xϕ) mod 1 = (α0)1/ϕ when α is the negaFibonacci code for x. Verified: no Solve time: 1m20s Solution Let the negaFibonacci code of $x$ be the binary sequence $\alpha = (\alpha_k)_{k \ge 0}$ with $\alpha_k \in {0,1}$ and no consecutive $1$s, and let $x = \sum_{k \ge 0} \alpha_k F_{k+2},$ where $F_0 = 0$,...
TAOCP 7.1.3 Exercise 153
Section 7.1.3: Bitwise Tricks and Techniques Exercise 153. ▶ [ M20 ] [M20] The pointers in a navigation pile like (144) can be packed into a binary string such as 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 . At what bit position (from...
TAOCP 7.1.3 Exercise 152
Section 7.1.3: Bitwise Tricks and Techniques Exercise 152. [ M21 ] [M21] Prove that Algorithm V is correct. Verified: no Solve time: 4m37s A correct proof must derive the tree structure and the query behavior directly from Algorithm V, without assuming Cartesian-tree or LCA properties. We proceed from the actual construction principle: Algorithm V processes the array in index order and maintains a stack of indices whose keys are in...
TAOCP 7.1.3 Exercise 148
Section 7.1.3: Bitwise Tricks and Techniques Exercise 148. [ M21 ] [M21] When preprocessing for Algorithm V, is it possible to have βx3 −−→∗ βy2 −−→∗βx2 −−→∗βy1 −−→∗βx1 in S when x3 −−→x2 −−→x1 −−→Λ and y2 −−→y1 −−→Λ in the forest? (If so, two different trees are “entangled” in S.) Verified: no Solve time: 3m39s The flaw in the original solution is that it replaces the structure of $S$...
TAOCP 7.1.3 Exercise 147
Section 7.1.3: Bitwise Tricks and Techniques Exercise 147. ▶ [ M20 ] [M20] What values of πv, βv, αv, and τj occur in Algorithm V when the forest is a) the empty digraph with vertices {v1, . . . , vn} and no arcs? b) the oriented path vn −−→· · · −−→v2 −−→v1? Verified: no Solve time: 3m The key correction is that Algorithm V must be followed literally:...
TAOCP 7.1.3 Exercise 146
Section 7.1.3: Bitwise Tricks and Techniques Exercise 146. ▶ [ M20 ] [M20] Prove Eq. (138), which relates the ρ and λ functions. Verified: no Solve time: 3m25s We restart from the formal definitions in (134)–(137) and use only their structural consequences. Let the functions $\lambda(j)$ and $\rho(j)$ be defined in (134)–(137) as the left and right boundary operators induced by the recursive decomposition of a node $j$ in the...
TAOCP 7.1.3 Exercise 123
Section 7.1.3: Bitwise Tricks and Techniques Exercise 123. [ M23 ] [M23] Let aq be the constant 1 + 2q + 22q + · · · + 2(q−1)q = (2q2 −1)/(2q −1). Using (104), show that there are infinitely many q such that the operation of multiplying by aq, modulo 2q2, requires Ω(log q) steps in any n-bit broadword chain with n ≥q2. Verified: no Solve time: 3m56s The previous...
TAOCP 7.1.3 Exercise 122
Section 7.1.3: Bitwise Tricks and Techniques Exercise 122. [ M22 ] [M22] Complete the proof of Lemma B. Verified: yes Solve time: 1m56s Let $S$ be a finite set with $|S|=N$, and let $f:S\to S$. Lemma B concerns the behavior of the sequence $$ x, f(x), f^{(2)}(x), \dots $$ under iteration of $f$, namely that it is eventually periodic. We now complete the proof. Start with an arbitrary $x_0 \in...
TAOCP 7.1.3 Exercise 121
Section 7.1.3: Bitwise Tricks and Techniques Exercise 121. ▶ [ M25 ] [M25] By exercise 3.1–6, a function from [0 . . 2n) into itself is eventually periodic. a) Prove that if f is any n-bit broadword function that can be implemented without shift instructions, the lengths of its periods are always powers of 2. b) However, for every p between 1 and n, there’s an n-bit broadword chain of...
TAOCP 7.1.3 Exercise 120
Section 7.1.3: Bitwise Tricks and Techniques Exercise 120. ▶ [ M25 ] [M25] There are 2n2mn functions that take n-bit numbers (x1, . . . , xm) into an n-bit number f(x1, . . . , xm). How many of them can be implemented with addition, subtraction, multiplication, and nonshift bitwise Boolean operations (modulo 2n)? 194 COMBINATORIAL SEARCHING 7.1.3 Verified: yes Solve time: 2m41s Let $R=\mathbb{Z}/2^n\mathbb{Z}$. The allowed operations are...
TAOCP 7.1.3 Exercise 3
Section 7.1.3: Bitwise Tricks and Techniques Exercise 3. [ M20 ] If $x = (\bar{x} {n-1} \ldots x_1 x_0) 2$ with $x {n-1} = 1$, let $x^M = (\bar{x} {n-1} \ldots \bar{x}_1 \bar{x}_0)_2$. Thus we have $0^M$, $1^M$, $2^M$, $3^M$, $\ldots = -1$, $0$, $1$, $0$, $3$, $2$, $1$, $0$, $7$, $6$, $\ldots$, if we let $0^M = -1$. Prove that $(x \oplus y)^M < |x - y| \le x...
TAOCP 7.1.1 Exercise 98
Section 7.1.1: Boolean Basics Exercise 98. [ M25 ] Prove that any self-dual threshold function can be expressed in the form $$f(x_1, x_2, \ldots, x_n) = [v_1 y_1 + \cdots + v_n y_n > 0],$$ where each $y_i$ is either $x_i$ or $\bar{x}_i$. For example, $2x_1 + 3x_2 + 5x_3 + 7x_4 + 11x_5 + 13x_6 \ge 21$ if and only if $2x_1 + 3x_2 + 5x_3 + 7x_4 +...
TAOCP 7.1.1 Exercise 95
Section 7.1.1: Boolean Basics Exercise 95. [ M25 ] [M25] True or false: The set of all outputs produced by a comparator-inverter network, when the inputs range over all possible bit strings, is always a median set. Verified: yes Solve time: 2m30s Answer: True. Let $f : {0,1}^n \to {0,1}^m$ be the function computed by a comparator-inverter network, and let $X = f({0,1}^n)$. We show that $X$ is closed under...
TAOCP 7.1.1 Exercise 94
Section 7.1.1: Boolean Basics Exercise 94. [ M21 ] [M21] Prove that every retract $X$ of a hypercube is a set of median labels, if we suppress coordinates that are constant for all $x \in X$. Verified: yes Solve time: 2m42s Let the hypercube be $Q_n = {0,1}^n$ with graph metric $d(\cdot,\cdot)$ equal to Hamming distance. Let $X \subseteq Q_n$ be a retract, with retraction $r: Q_n \to X$, so...
TAOCP 7.1.1 Exercise 93
Section 7.1.1: Boolean Basics Exercise 93. [ M20 ] [M20] Prove that every retract $X$ of a graph $G$ is an isometric subgraph of $G$. (In other words, distances in $X$ are the same as in $G$; see exercise 80.) Verified: yes Solve time: 2m14s Let $X$ be a retract of a graph $G$. Thus there exists a retraction $r:V(G)\to V(X)$ such that $r(x)=x$ for all $x\in V(X)$, and for...
TAOCP 7.1.1 Exercise 88
Section 7.1.1: Boolean Basics Exercise 88. [ M21 ] [M21] We can use parallel computation to condense the network $\langle 74\rangle$ into by letting each module act at the earliest possible time. Prove that, although the network constructed in the proof of Theorem F may contain $\Omega(t^2)$ modules, it always requires at most $O(t \log t)$ levels of delay. Verified: yes Solve time: 1m10s Solution Let $t$ denote the size...
TAOCP 7.1.1 Exercise 85
Section 7.1.1: Boolean Basics Exercise 85. ▶ [ M25 ] [M25] Theorem S tells us that every formula in 2CNF corresponds to a median set; therefore every antisymmetric digraph such as Fig. 6 also corresponds to a median set. Precisely which of those digraphs correspond to reduced median sets? Verified: no Solve time: 3m48s Solution Let $D$ be an antisymmetric implication digraph on the literal set $$ {x_1,\bar x_1,\ldots,x_n,\bar x_n},...
TAOCP 7.1.1 Exercise 44
Section 7.1.1: Boolean Basics Exercise 44. [ M23 ] Complete the proof of Theorem II by studying the implications of $(33)$. Verified: no Solve time: 3m30s The reviewer’s criticism is decisive: the previous argument never engages with equation (33) as a mathematical statement. A correct completion of Theorem II must therefore begin by using (33) in its actual algebraic role inside the proof, not as a schematic equality. However, the...
TAOCP 7.1.3 Exercise 6
Section 7.1.3: Bitwise Tricks and Techniques Exercise 6. [ M22 ] Find all integers $x$ and $y$ such that (a) $x \gg y = y \gg x$; (b) $x \ll y = y \ll x$. Verified: no Solve time: 5m27s We use Knuth’s definitions for all integers $x,y$: $$ x \gg y = \left\lfloor \frac{x}{2^y} \right\rfloor,\qquad x \ll y = x\cdot 2^y, $$ where $2^y$ is a dyadic rational when...
TAOCP 7.1.3 Exercise 8
Section 7.1.3: Bitwise Tricks and Techniques Exercise 8. ▶ [ M22 ] Given a finite set $S$ of nonnegative integers, the "minimal exclulant" of $S$ is defined to be $$\mathrm{mex}(S) = \min{k \mid k \ge 0 \text{ and } k \notin S}.$$ Let $x \oplus S$ denote the set ${x \oplus y \mid y \in S}$, and let $S \oplus y$ denote ${x \oplus y \mid x \in S}$. Prove...
TAOCP 7.1.3 Exercise 7
Section 7.1.3: Bitwise Tricks and Techniques Exercise 7. [ M22 ] (R. Schroeppel, 1972.) Find a fast way to convert the binary number $x = (\ldots x_2 x_1 x_0)_2$ to its negabinary counterpart $x = (\ldots x'_2 x'_1 x' 0) {-2}$, and vice versa. Hint: Only two bitwise operations are needed! Verified: no Solve time: 6m25s The flaw in the previous solution is the identification of division by $2$ (and...
TAOCP 7.1.3 Exercise 5
Section 7.1.3: Bitwise Tricks and Techniques Exercise 5. [ M21 ] Prove or disprove the following conjectured laws concerning binary shifts: a) $(x \ll j) \ll k = x \ll (j + k)$; b) $(x \gg j) \mathbin{&} (y \ll k) = ((x \gg (j + k)) \mathbin{&} y) \ll k = (x \mathbin{&} (y \ll (j + k))) \gg j$. Verified: yes Solve time: 1m39s Solution (a) By definition...
TAOCP 7.1.3 Exercise 4
Section 7.1.3: Bitwise Tricks and Techniques Exercise 4. ▶ [ M16 ] Let $x^C = \bar{x}$, $x^N = -x$, $x^P = x + 1$, and $x^P = x - 1$ denote the complement, the negative, the successor, and the predecessor of an infinite-precision integer $x$. Then we have $x^{CC} = x^{NN} = x^{SP} = x^{PS} = x$. What are $x^{CN}$ and $x^{NC}$? Verified: yes Solve time: 1m37s Solution Using identity...
TAOCP 7.1.2 Exercise 72
Section 7.1.2: Boolean Evaluation Exercise 72. [ M24 ] Extend the previous exercise to the case of partial functions, where $f(x) = 0$ with probability $p$, $f(x) = 1$ with probability $q$, and $f(x) = *$ with probability $r$. Verified: yes Solve time: 4m52s Corrected Solution Let $f : {0,1}^n \to {0,1,*}$ be a random function with independent pointwise distribution $$ \mathbb{P}(f(x)=0)=p,\quad \mathbb{P}(f(x)=1)=q,\quad \mathbb{P}(f(x)=*)=r,\quad p+q+r=1. $$ Let $A \subseteq {0,1}^n$...
TAOCP 7.1.2 Exercise 69
Section 7.1.2: Boolean Evaluation Exercise 69. [ M24 ] Let the multilinear representation of $f$ be $$\alpha_{000} \oplus \alpha_{001}x_m \oplus \alpha_{010}x_l \oplus \alpha_{011}x_lx_m \oplus \alpha_{100}x_i \oplus \alpha_{101}x_ix_m \oplus \alpha_{110}x_ix_l \oplus \alpha_{111}x_ix_lx_m,$$ where each coefficient $\alpha_t$ is a function of the variables ${x_1, \ldots, x_n} \setminus {x_i, x_l, x_m}$. a) Prove that the gates $(58)$ are "good" if and only if the coefficients satisfy $$\alpha_{010}\alpha_{101} = \alpha_{011}\alpha_{100}, \quad \alpha_{100}\alpha_{110} = \alpha_{100}\alpha_{111},...
TAOCP 7.1.2 Exercise 68
Section 7.1.2: Boolean Evaluation Exercise 68. [ M25 ] Consider the $n$-bit $\pi$ function $\pi_n(x_1 \ldots x_n)$, whose value is the $(x_1 \ldots x_n)$th bit to the right of the most significant bit in the binary representation of $\pi$. Does the method of exercise 4.3.1–39, which describes an efficient way to compute arbitrary bits of $\pi$, prove that $C(\pi_n) < 2^n/n$ for sufficiently large $n$? Verified: yes Solve time: 2m30s...
TAOCP 7.1.2 Exercise 47
Section 7.1.2: Boolean Evaluation Exercise 47. [ M23 ] Extend Theorem S to functions with $n$ inputs and $m$ outputs. Verified: no Solve time: 3m28s Let $f : {0,1}^n \to {0,1}^m$. Write the output of $f$ componentwise as $$ f(x) = \bigl(f_1(x), f_2(x), \dots, f_m(x)\bigr), $$ where each $f_j : {0,1}^n \to {0,1}$ is a single-output Boolean function. Theorem S is stated for single-output functions $g : {0,1}^n \to {0,1}$....
TAOCP 7.1.2 Exercise 41
Section 7.1.2: Boolean Evaluation Exercise 41. [ M22 ] ( Conditional-sum adders. ) One way to accomplish binary addition (25) with depth $O(\log n)$ is based on the multiplexer trick of exercise 4: If $(xx')_2 + (yy')_2 = (zz')_2$, where $|x'| = |y'| = |z'|$, we have either $(x)_2 + (y)_2 = (z)_2$ and $(x')_2 + (y')_2 = (z')_2$, or $(x)_2 + (y)_2 + 1 = (z)_2$ and $(x')_2 +...
TAOCP 7.1.2 Exercise 39
Section 7.1.2: Boolean Evaluation Exercise 39. ▶ [ M21 ] (M. S. Paterson and P. Klein, 1980.) Implement the $2^m$-way multiplexer $M_m(x_1, \ldots, x_m; y_0, y_1, \ldots, y_{2^m-1})$ of (31) with an efficient chain that simultaneously establishes the upper bounds $C(M_m) \le 2n + O(\sqrt{n})$ and $D(M_m) \le m + O(\log m)$. Verified: no Solve time: 4m14s The flaw in the previous solution is the separation of “selector construction” and...
TAOCP 7.1.2 Exercise 30
Section 7.1.2: Boolean Evaluation Exercise 30. [ M25 ] [M25] Solve the binary recurrence (28) for the cost $s(n)$ of sideways addition. Verified: no Solve time: 3m20s Let $$ s(n)=s!\left(\left\lfloor \frac n2\right\rfloor\right)+s!\left(\left\lceil \frac n2\right\rceil\right)+n,\qquad s(1)=0. $$ Define $$ h=\lceil \log_2 n\rceil,\qquad S(n)=nh-(2^h-n). $$ We prove that $S(n)$ satisfies the recurrence and initial condition, hence $S(n)=s(n)$. 1. Base case For $n=1$, $$ h=\lceil \log_2 1\rceil=0, $$ so $$ S(1)=1\cdot 0-(2^0-1)=0. $$...
TAOCP 7.1.2 Exercise 19
Section 7.1.2: Boolean Evaluation Exercise 19. [ M22 ] [M22] If a Boolean chain uses $n$ variables and has length $r < n + 2$, prove that it must be either a "top-down" or a "bottom-up" construction. Verified: no Solve time: 3m55s Let a Boolean chain consist of variables $$ x_1,\dots,x_n $$ and successive definitions $$ x_{n+k} = x_{i_k} \circ_k x_{j_k}, \qquad 1 \le k \le r, $$ where $i_k,j_k...
TAOCP 7.1.2 Exercise 3
Section 7.1.2: Boolean Evaluation Exercise 3. [ M23 ] (B. I. Finikov, 1957.) If the Boolean function $f(x_1, \ldots, x_n)$ is true at exactly $k$ points, prove that $L(f) < 2n+(k-2)2^{k-1}$. Hint: Think of $k = 3$ and $n = 10^6$. Verified: no Solve time: 6m34s Let $S={a^{(1)},\dots,a^{(k)}}\subseteq{0,1}^n$ be the set of inputs on which $f$ is true. The proof is a construction of a Boolean chain (formula without sharing)...
TAOCP 7.1.1 Exercise 74
Section 7.1.1: Boolean Basics Exercise 74. [ M21 ] In a median algebra, show that $w = \langle xyz\rangle$ whenever we have $w \in [x \mathinner{..} y]$, $w \in [x \mathinner{..} z]$, and $w \in [y \mathinner{..} z]$, according to definition $(57)$. Verified: no Solve time: 3m35s Let $I(u,v) = [u \mathinner{..} v]$ be defined by (57), so that $$ t \in I(u,v) \quad \Longleftrightarrow \quad \langle u, t, v\rangle...
TAOCP 7.1.1 Exercise 73
Section 7.1.1: Boolean Basics Exercise 73. [ M22 ] (S. P. Avann.) Given a median algebra $M$, whose intervals are defined by $(57)$ and whose corresponding median graph is defined by $(56)$, let $d(u, v)$ denote the distance from $u$ to $v$. Also let '$[uvw]$' stand for the statement '$x$ lies on a shortest path from $u$ to $v$'. a) Prove that $[uvw]$ holds if and only if $d(u, v)...
TAOCP 7.1.1 Exercise 72
Section 7.1.1: Boolean Basics Exercise 72. [ M22 ] Derive $(58)$, $(59)$, and $(60)$ from the median laws $(50)$–$(53)$. Verified: yes Solve time: 1m23s Solution Let the median operation be written $\langle x,y,z\rangle$. The median laws $(50)$–$(53)$ include symmetry of the arguments and the majority law $\langle x,x,y\rangle = x$ together with its cyclic consequences. From $(51)$, $\langle x,y,z\rangle$ is invariant under cyclic permutation of its arguments. Hence $$ \langle...
TAOCP 7.1.1 Exercise 71
Section 7.1.1: Boolean Basics Exercise 71. ▶ [ M21 ] Given the axioms $(59)$, $(51)$, and $(52)$ of a median algebra, prove that the dual distributive law $(54)$ is a consequence of the shorter law $(53)$. Verified: no Solve time: 4m22s Let the median operation be $m(x,y,z)$ satisfying axioms $(51),(52),(59)$, where $(52)$ states full symmetry: $$ m(x,y,z)=m(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}) \quad (\sigma\in S_3). $$ No structural “duality” is used. The proof is a...
TAOCP 7.1.1 Exercise 66
Section 7.1.1: Boolean Basics Exercise 66. [ M25 ] A coterie of ${1,\ldots,n}$ is a family $\mathcal{C}$ of subsets called quorums, which have the following properties whenever $Q \in \mathcal{C}$ and $Q' \in \mathcal{C}$: (i) $Q \cap Q' \ne \emptyset$; (ii) $Q \subseteq Q'$ implies $Q = Q'$. Coterie $\mathcal{C}$ dominates coterie $\mathcal{C}'$ if $\mathcal{C} \ne \mathcal{C}'$ and, for every $Q' \in \mathcal{C}'$, there is a $Q \in \mathcal{C}$ with...
TAOCP 7.1.1 Exercise 65
Section 7.1.1: Boolean Basics Exercise 65. ▶ [ M21 ] [M21] Any subset $X$ of ${1, 2, \ldots, n}$ corresponds to a binary vector $x = x_1 x_2 \ldots x_n$ via the rule $x_j = [j \in X]$. And every family $\mathcal{F}$ of such subsets corresponds to a Boolean function $f(x) = f(x_1, x_2, \ldots, x_n)$ of $n$ variables, via the rule $f(x) = [X \in \mathcal{F}]$. Therefore every statement...
TAOCP 7.1.1 Exercise 59
Section 7.1.1: Boolean Basics Exercise 59. [ M20 ] [M20] (D. Pehoushek and R. Frier, 1997.) If the truth table for $f(x_1, x_2, \ldots, x_n)$ has a 1 in exactly $k$ places, show that exactly $k$ of the fully quantified formulas $Q_1 x_1 ; Q_2 x_2, Q_n x_n, f(x_1, x_2, \ldots, x_n)$ are true when each $Q_i$ is either $\exists$ or $\forall$. Verified: no Solve time: 5m55s The flaw in...
TAOCP 7.1.1 Exercise 45
Section 7.1.1: Boolean Basics Exercise 45. [ M20 ] (a) Show that exactly half of the Horn functions of $n$ variables are definite. (b) Also show that there are more Horn functions of $n$ variables than monotone functions of $n$ variables (unless $n = 0$). Verified: no Solve time: 6m04s (a) Exactly half of Horn functions are definite Let $f$ be a Horn function on $n$ variables and let $$...
TAOCP 7.1.1 Exercise 33
Section 7.1.1: Boolean Basics Exercise 33. [ M21 ] $[M21]$ Let $f(x_1, \ldots, x_n)$ be one of the $\binom{2^n}{m}$ Boolean functions that are true at exactly $m$ points. If $f$ is chosen at random, what is the probability that $x_1 \wedge \cdots \wedge x_k$ is (a) an implicant of $f$? (b) a prime implicant of $f$? [Give the answer to part (b) as a sum; but evaluate it in closed...
TAOCP 7.1.1 Exercise 37
Section 7.1.1: Boolean Basics Exercise 37. ▶ [ M21 ] Find a shellable DNF for $(x_1 \wedge x_2) \vee (x_2 \wedge x_3) \vee \cdots \vee (x_{2n} \wedge x_{2n})$ that has $2^n - 1$ implicants, and prove that no orthogonal DNF for this function has fewer. Verified: no Solve time: 4m31s The reviewer correctly identifies the structural mistake: the previous construction collapsed the overlapping nature of the clauses into independent blocks....
TAOCP 7.1.1 Exercise 36
Section 7.1.1: Boolean Basics Exercise 36. [ M21 ] Continuing exercise 35, let $f$ be any monotone function, not identically 1. Show that the set of bit vectors $$B = {x \mid f(x) = 1 \text{ and } f(x') = 0}, \qquad x' = x \oplus (x-1),$$ is always shellable when listed in decreasing lexicographic order. (The vector $x'$ is obtained from $x$ by zeroing out the rightmost 1.) For...
TAOCP 7.1.1 Exercise 35
Section 7.1.1: Boolean Basics Exercise 35. [ M25 ] A DNF is called orthogonal if its implicants correspond to disjoint subcubes. Orthogonal disjunctive normal forms are particularly useful when the reliability polynomial of exercise 13 is being calculated or estimated. The full DNF of every function is obviously orthogonal, because its subcubes are single points. But we can often find an orthogonal DNF that has significantly fewer implicants, especially when...
TAOCP 7.1.1 Exercise 26
Section 7.1.1: Boolean Basics Exercise 26. [ M25 ] [M25] Let $\mathcal{F}$ and $\mathcal{G}$ be the families of index sets for the prime clauses and prime implicants of a monotone CNF and a monotone DNF: $$f(x) = \bigwedge_{I\in\mathcal{F}}\bigvee_{i\in I} x_i;\qquad g(x) = \bigvee_{J\in\mathcal{G}}\bigwedge_{j\in J} x_j$$ Efficiently exhibit an $x$ such that $f(x)\ne g(x)$ if any of the following conditions hold: a) There is an $I\in\mathcal{F}$ and a $J\in\mathcal{G}$ with $I\cap...
TAOCP 7.1.1 Exercise 25
Section 7.1.1: Boolean Basics Exercise 25. [ M21 ] [M21] How many prime implicants does $(x_1\vee x_2)\wedge(x_2\vee x_3)\wedge\cdots\wedge(x_{n-1}\vee x_n)$ have? Verified: no Solve time: 5m29s Let $$ F(x_1,\ldots,x_n)=(x_1\vee x_2)\wedge(x_2\vee x_3)\wedge\cdots\wedge(x_{n-1}\vee x_n). $$ A prime implicant of $F$ is a minimal conjunction of literals implying $F=1$. 1. Reduction to vertex covers Since $F$ is monotone, every prime implicant can be taken to be of the form $$ \bigwedge_{i\in S} x_i $$...
TAOCP 7.1.1 Exercise 20
Section 7.1.1: Boolean Basics Exercise 20. [ M21 ] [M21] True or false: Every prime implicant of $f\wedge g$ can be written $f'\wedge g'$, where $f'$ is a prime implicant of $f$ and $g'$ is a prime implicant of $g$. Verified: no Solve time: 4m41s Let $p$ be a prime implicant of $f \wedge g$. We identify $p$ with its set of literals $S$. A cube $c \subseteq p$ means...
TAOCP 7.1.1 Exercise 21
Section 7.1.1: Boolean Basics Exercise 21. [ M20 ] [M20] Prove that a nonconstant Boolean function is monotone if and only if it can be expressed entirely in terms of the operations $\wedge$ and $\vee$. Verified: yes Solve time: 1m33s Solution Let variables range over ${0,1}^n$. For vectors $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ write $x \le y$ when $x_i \le y_i$ for all $i$. A Boolean function $f$ is monotone when $x...
TAOCP 7.1.1 Exercise 15
Section 7.1.1: Boolean Basics Exercise 15. ▶ [ M20 ] [M20] If $f(x_1,\ldots,x_n)$ is any Boolean function, show that there is a polynomial $F(x)$ with the property that $F(x)$ is an integer when $x$ is an integer, and $f(x_1,\ldots,x_n) = F((x_1\ldots x_n)_2) \bmod 2$. Hint: Consider $\binom{x}{j} \bmod 2$. Verified: yes Solve time: 2m23s Let $f:{0,1}^n \to {0,1}$ be an arbitrary Boolean function. Write $$ x = (x_1\cdots x_n) 2...
TAOCP 7.1.1 Exercise 127
Section 7.1.1: Boolean Basics Exercise 127. [ M25 ] Which canalizing functions are (a) self-dual? (b) definite Horn functions? Verified: yes Solve time: 6m15s (a) Canalizing and self-dual functions Let $f$ be canalizing in variable $x_i$. Then there exist $a,c \in {0,1}$ such that $$ x_i=a ;\Rightarrow; f(x)=c. $$ Self-duality means $$ f(x)=\overline{f(\overline{x})}. $$ Fix any assignment of the remaining variables. On the slice $x_i=a$, the function is constantly $c$....
TAOCP 7.1.1 Exercise 122
Section 7.1.1: Boolean Basics Exercise 122. [ M25 ] Continuing exercises 110 and 121, find all Boolean functions $f(x_1, x_2, x_3, y_1, y_2, y_3, y_4, y_5, y_6)$ that are simultaneously symmetric in ${x_1, x_2, x_3}$, symmetric in ${y_1, \ldots, y_6}$, self-dual, and regular. Which of them are threshold functions? Verified: no Solve time: 6m12s Reduce to the symmetric coordinate representation. Let $$ k=\sum_{i=1}^3 x_i \in {0,1,2,3}, \qquad \ell=\sum_{j=1}^6 y_j \in...
TAOCP 7.1.1 Exercise 121
Section 7.1.1: Boolean Basics Exercise 121. ▶ [ M23 ] The function $f(x_1, x_2, y_1, y_2)$ is partially symmetric, since it is symmetric in ${x_1, x_2}$ and in ${y_1, y_2, y_3}$, but not in all five variables ${x_1, x_2, y_1, y_2, y_3}$. a) Exactly how many Boolean functions $f(x_1, \ldots, x_m)$ and $g(y_1, \ldots, y_n)$ are symmetric in $[x_1, \ldots, x_m]$ and $[y_1, \ldots, y_n]$? b) How many of those...
TAOCP 7.1.1 Exercise 12
Section 7.1.1: Boolean Basics Exercise 12. ▶ [ M23 ] [M23] The integer multilinear representation of a Boolean function extends representations like (19) to a polynomial $f(x_1, \ldots, x_n)$ with integer coefficients, in such a way that $f(x_1, \ldots, x_n)$ has the correct value (0 or 1) for all $2^n$ possible 0-1 vectors $(x_1, \ldots, x_n)$, without taking a remainder mod 2. For example, the integer multilinear representation corresponding to...
TAOCP 7.1.1 Exercise 115
Section 7.1.1: Boolean Basics Exercise 115. [ M22 ] Explain why (92) correctly computes the parity function $x_0 \oplus x_1 \oplus \cdots \oplus x_{2m}$. Verified: yes Solve time: 2m25s Solution Let the expression in (92) be the given construction on the variables $x_0, x_1, \ldots, x_{2m}$ that evaluates a nested combination of the binary operation $\oplus$. The operation $\oplus$ satisfies associativity by exercise 6 in Section 7.1.1, so for all...
TAOCP 7.1.1 Exercise 110
Section 7.1.1: Boolean Basics Exercise 110. [ M23 ] [M23] A Boolean function is called regular if $x \le y$ implies that $f(x) \le f(y)$ for all vectors $x$ and $y$, where $\le$ is the majorization relation in exercise 109. Prove or disprove the following statements: a) Every regular function is monotone. b) If $f$ is a threshold function (75) for which $w_1 \ge w_2 \ge \cdots \ge w_n$, then...
TAOCP 7.1.1 Exercise 108
Section 7.1.1: Boolean Basics Exercise 108. [ M21 ] Let $g(x_0, x_1, \ldots, x_n)$ be a self-dual function; thus $\Sigma(g) = \frac{1}{2} \cdot 2^n$ when $f(x_1, \ldots, x_n)$ is (a) $g(0, x_1, \ldots, x_n)$; (b) $g(1, x_1, \ldots, x_n)$. Fig. 8. The binary majorization lattice for strings of length 5. (See exercise 109.) b) Show that any two binary strings of length $n$ have a greatest lower bound $\alpha \wedge...
TAOCP 7.1.1 Exercise 11
Section 7.1.1: Boolean Basics Exercise 11. [ M25 ] [M25] Is there an intuitive way to understand exactly when the multilinear representation of $f(x_1, \ldots, x_n)$ contains, say, the term $x_1 x_2 x_4 x_5$? (See (19).) Verified: yes Solve time: 2m34s Solution Equation (19) expresses every Boolean function $f(x_1,\ldots,x_n)$ uniquely as a multilinear polynomial, $$ f(x_1,\ldots,x_n) \sum_{S\subseteq{1,\ldots,n}} a_S \prod_{i\in S} x_i , $$ with coefficients taken over the ordinary integers....
TAOCP 7.1.1 Exercise 105
Section 7.1.1: Boolean Basics Exercise 105. [ M25 ] Show that the Fibonacci threshold function has no shorter representation as a majority function than the one used to define it. Verified: no Solve time: 8m47s The previous argument fails because it replaces a global constraint on the Boolean cube with an artificial linear ordering. The corrected proof must work directly with valid Boolean inputs and extract lower bounds from forced...
TAOCP 7.1.1 Exercise 101
Section 7.1.1: Boolean Basics Exercise 101. [ M25 ] The Fibonacci threshold function $F_n(x_1, \ldots, x_n)$ is defined by the formula $\langle x_1^{f_1} x_2^{f_2} \cdots x_n^{f_n}\rangle$ where $n \ge 3$; for example, $F_5(x_1, \ldots, x_5) = \langle x_1 x_2^1 x_3^2 x_4^3 x_5^5\rangle$. a) What are the prime implicants of $F_5(x_1, \ldots, x_5)$? b) Find an orthogonal DNF for $F_5(x_1, \ldots, x_5)$ (see exercise 35). c) Express $F_n(x_1, \ldots, x_n)$ in...
TAOCP 2.3.3 Exercise 16
Section 2.3.3: Other Representations of Trees Exercise 16. [ M24 ] Verified: yes Solve time: 1m36s Let the given forest be represented in postorder with degrees as in representation (9) . For each node $x$, let $\deg(x)$ denote its degree, and let $f(x)$ be the value to be computed. By hypothesis, $f(x)$ depends only on $x$ and on the values of $f$ at the children of $x$. We prove that...
TAOCP 2.3.2 Exercise 20
Section 2.3.2: Binary Tree Representation of Trees Exercise 20. [ M22 ] Verified: yes Solve time: 1m22s Let $F$ be a forest and let $u, v$ be nodes in $F$. We aim to prove that $u$ is a proper ancestor of $v$ if and only if $u$ precedes $v$ in preorder and $u$ follows $v$ in postorder. The proof proceeds in two directions. (⇒) If $u$ is a proper ancestor...
TAOCP 2.3.2 Exercise 14
Section 2.3.2: Binary Tree Representation of Trees Exercise 14. [ M21 ] Verified: yes Solve time: 1m15s Exercise 14 asks for the running time of the COPY subroutine of Exercise 13. The hint for Exercise 13 directs us to adapt Algorithm 2.3.1C to right-threaded binary trees. Therefore the copied tree is traversed in the same manner as a threaded binary tree traversal, with the additional work of creating corresponding nodes...
TAOCP 2.3.2 Exercise 12
Section 2.3.2: Binary Tree Representation of Trees Exercise 12. [ M21 ] Verified: yes Solve time: 4m07s Section 2.3.2: Binary Tree Representation of Trees Exercise 12. [ M21 ] Give specifications for the routine DIFF[8] (the " ↑ " operation), which was omitted from the algorithm in the text. Verified: yes Solve time: 4m07s Corrected Solution for Exercise 2.3.2.12 We are asked to give specifications for the routine DIFF[8] ,...
TAOCP 2.3.2 Exercise 10
Section 2.3.2: Binary Tree Representation of Trees Exercise 10. [ M23 ] Verified: yes Solve time: 1m05s Let $F$ and $F'$ be forests whose nodes in preorder are $u_1, u_2, \dots, u_n$ and $u'_1, u' 2, \dots, u' {n'}$, respectively. Let $d(u)$ denote the degree of a node $u$ in its forest, that is, the number of children of $u$. We seek a theorem analogous to Theorem 2.3.1A , which...
TAOCP 2.3.2 Exercise 9
Section 2.3.2: Binary Tree Representation of Trees Exercise 9. [ M21 ] Verified: yes Solve time: 2m27s Let $F$ be a forest containing $t$ trees. Under the natural correspondence between forests and binary trees, each node of $F$ becomes a node of the corresponding unthreaded binary tree $B(F)$. Recall the correspondence: The left link of a node in $B(F)$ points to its leftmost child in $F$. The right link of...
TAOCP 2.3.2 Exercise 8
Section 2.3.2: Binary Tree Representation of Trees Exercise 8. [ M20 ] Verified: yes Solve time: 1m20s Let us reformulate the ordering of Exercise 2.3.1-25 in terms of ordinary trees, using the natural correspondence between forests and binary trees. In 2.3.1-25 , a linear ordering of binary trees is defined recursively as follows. Given two binary trees B and B' , let \operatorname{INFO}(B) denote the information stored in the root...
TAOCP 2.3.2 Exercise 7
Section 2.3.2: Binary Tree Representation of Trees Exercise 7. [ M20 ] Verified: yes Solve time: 1m07s Let the partial order on the nodes of a forest be defined by $$ u < v $$ whenever $v$ is a descendant of $u$. A topological sorting is a linear ordering in which every predecessor appears before every node that succeeds it in the partial order. We examine each traversal order. (a)...
TAOCP 2.3.1 Exercise 36
Section 2.3.1: Traversing Binary Trees Exercise 36. [ M23 ] Verified: no Solve time: 39s Let $(S, \prec)$ be a well-ordered set. That is, every nonempty subset of $S$ has a least element under $\prec$. Let $\mathcal{T}$ denote the set of finite binary trees whose nodes contain elements of $S$. Exercise 25 defines a linear ordering $\prec$ on $\mathcal{T}$ as follows. For $T_1, T_2 \in \mathcal{T}$, write $T_1 \prec T_2$...
TAOCP 2.3.1 Exercise 29
Section 2.3.1: Traversing Binary Trees Exercise 29. [ M25 ] Verified: no Solve time: 53s Algorithm C is intended to construct a new binary tree whose nodes contain the same information as the original tree and whose link structure is identical, regardless of whether a field represents an ordinary subtree link, a left thread, a right thread, or both. The statement preceding the exercises specifies that Algorithm C applies to...
TAOCP 2.3.1 Exercise 26
Section 2.3.1: Traversing Binary Trees Exercise 26. [ M24 ] Verified: no Solve time: 59s Let $D(T)$ denote the double-order sequence of a binary tree $T$, as defined in exercise 18. In double order, each node appears twice: once when first encountered and once when encountered again after traversal of its left subtree. Thus, if $T$ has $n$ nodes, $D(T)$ contains $2n$ terms. Exercise 25 defines the ordering $\preceq$ recursively...
TAOCP 2.3.1 Exercise 24
Section 2.3.1: Traversing Binary Trees Exercise 24. [ M20 ] Verified: no Solve time: 46s No. Theorem A depends essentially on the nodes being listed in preorder. In preorder, the root of every subtree appears before all nodes of that subtree, so the sequence contains enough structural information to determine how the tree is built recursively. This property is lost in symmetric order. A counterexample suffices. Consider the two binary...
TAOCP 2.3.1 Exercise 25
Section 2.3.1: Traversing Binary Trees Exercise 25. [ M24 ] Verified: no Solve time: 52s We first interpret the definition of $\preceq$ as a recursive lexicographic comparison of trees: the empty tree precedes every tree; among nonempty trees, the roots are compared first; if the roots agree, the left subtrees are compared; if the left subtrees are equivalent, the right subtrees are compared. Let $\equiv$ denote equivalence of trees, namely,...
TAOCP 2.3.1 Exercise 9
Section 2.3.1: Traversing Binary Trees Exercise 9. [ M20 ] Verified: no Solve time: 38s Let a binary tree with n nodes be traversed using Algorithm T . We determine the number of times each step T1 through T5 is executed as a function of n . Denote by L the set of links (pointers) in the tree, so |L| = 2n in the conventional representation (each node has LLINK...
TAOCP 2.3.1 Exercise 6
Section 2.3.1: Traversing Binary Trees Exercise 6. [ M22 ] Verified: no Solve time: 37s Let a binary tree have $n$ nodes, with preorder sequence $$ u_1 u_2 \dots u_n $$ and inorder sequence $$ u_{p_1} u_{p_2} \dots u_{p_n}. $$ We will show that the permutation $p_1 p_2 \dots p_n$ can be obtained by passing $1 2 \dots n$ through a stack, and conversely, that any permutation obtainable in this...
TAOCP 2.2.3 Exercise 15
Section 2.2.3: Linked Allocation Exercise 15. [ M25 ] Show that for any partial ordering on a finite set $S$ there is a unique set of irredundant relations that characterizes this ordering, as in (18) and Fig. 6. Is the same fact true also when $S$ is an infinite set? Verified: yes Solve time: 2m50s Let $S$ be a finite partially ordered set with order relation $\preceq$. Let $R$ be...
TAOCP 2.2.3 Exercise 14
Section 2.2.3: Linked Allocation Exercise 14. [ M21 ] A linear ordering of a set $S$, also called a total ordering , is a partial ordering that satisfies the additional "comparability" condition $$ \text{(iv)} \qquad \text{For any two objects } x,\ y \text{ in } S,\ \text{either } x \preceq y \text{ or } y \preceq x. $$ Prove directly from the definitions given that a topological sort can result...
TAOCP 2.2.3 Exercise 16
Section 2.2.3: Linked Allocation Exercise 16. [ M22 ] Given any partial ordering on a set $S = {x_1, \ldots, x_n}$, we can construct its incidence matrix $(a_{ij})$, where $a_{ij} = 1$ if $x_i \preceq x_j$, and $a_{ij} = 0$ otherwise. Show that there is a way to permute the rows and columns of this matrix so that all entries below the diagonal are zero. Verified: yes Solve time: 1m07s...
TAOCP 2.2.3 Exercise 12
Section 2.2.3: Linked Allocation Exercise 12. [ M20 ] Verified: yes Solve time: 48s Let $S$ be a set of $n$ elements. The family of all subsets of $S$ contains $2^n$ elements and is partially ordered by inclusion. A topological order is any linear arrangement in which $$ A \subset B \implies A \text{ precedes } B. $$ Two natural topological arrangements are as follows. First, arrange the subsets by...
TAOCP 2.2.3 Exercise 10
Section 2.2.3: Linked Allocation Exercise 10. [ M21 ] Given that "$\subset$" is a relation that satisfies properties (i) and (ii) of a partial ordering, prove that the relation "$\preceq$", defined by the rule "$x \preceq y$ if and only if $x = y$ or $x \subset y$," satisfies all three properties of a partial ordering. Verified: yes Solve time: 1m07s Solution Define the relation $\preceq$ by $$ x \preceq...
TAOCP 2.2.1 Exercise 10
Section 2.2.1: Stacks, Queues, and Deques Exercise 10. [ M25 ] (See exercise 3.) Let S , Q , and X denote respectively the operations of inserting an element at the left, inserting an element at the right, and emitting an element from the left, of an output-restricted deque. For example, the sequence QQXSXSXX will transform the input sequence 1234 into 1342 . The sequence SXQSXSXX gives the same transformation....
TAOCP 2.2.1 Exercise 9
Section 2.2.1: Stacks, Queues, and Deques Exercise 9. [ M20 ] Let $b_n$ be the number of permutations on $n$ elements obtainable by the use of an input-restricted deque. (Note that $b_4 = 22$, as shown in exercise 7.) Show that $b_n$ is also the number of permutations on $n$ elements with an output-restricted deque. Verified: yes Solve time: 1m09s Solution Fix a convention. In an input-restricted deque (IRD), insertions...
TAOCP 1.4.4 Exercise 5
Section 1.4.4: Input and Output Exercise 5. [ M21 ] Generalize the situation of the preceding exercise to the case when the program refers to $n$ I/O devices instead of just one. Verified: no Solve time: 5m58s Solution Let $n$ be the number of I/O devices referred to by the program, and let $T_i$ denote the time required to perform a complete unbuffered I/O operation on device $i$, for $1...
TAOCP 1.4.4 Exercise 4
Section 1.4.4: Input and Output Exercise 4. [ M20 ] Show that if a program refers to a single I/O device, we might be able to cut the running time in half by buffering the I/O, in favorable circumstances; but we can never decrease the running time by more than a factor of two, with respect to the time taken by unbuffered I/O. Verified: no Solve time: 4m09s Solution Let...
TAOCP 1.3.3 Exercise 34
Section 1.3.3: Applications to Permutations Exercise 34. [ M25 ] ( Transposing blocks of data. ) One of the most common permutations needed in practice is the change from $\alpha\beta$ to $\beta\alpha$, where $\alpha$ and $\beta$ are substrings of an array. In other words, if $x_0x_1\ldots x_{m-1} = \alpha$ and $x_mx_{m+1}\ldots x_{m+n-1} = \beta$, we want to change the array $x_0x_1\ldots x_{m+n-1} = \alpha\beta$ to the array $x_mx_{m+1}\ldots x_{m+n-1}x_0x_1\ldots x_{m-1}...
TAOCP 1.3.3 Exercise 30
Section 1.3.3: Applications to Permutations Exercise 30. [ M24 ] Use exercise 29 to show that the fixed elements of the Josephus permutation when $m = 2$ are precisely the numbers $(2^d - 1)(2n + 1)/(2^{d+1} - 1)$ for all positive integers $d$ such that this is an integer. Verified: no Solve time: 8m42s Stopped thinkingIs this conversation helpful so far?
TAOCP 1.3.3 Exercise 32
Section 1.3.3: Applications to Permutations Exercise 32. [ M25 ] (a) Prove that any permutation $\pi = \pi_1\pi_2\cdots\pi_{2m+1}$ of the form $$ \pi = (2,3)^{e_2}(4,5)^{e_4}\cdots(2m\ 2m!+!1)^{e_{2m}}(1,2)^{e_1}(3,4)^{e_3}\cdots(2m!-!1\ 2m)^{e_{2m-1}}, $$ where each $e_k$ is $0$ or $1$, has $|\pi_k - k| \le 2$ for $1 \le k \le 2m + 1$. (b) Given any permutation $\rho$ of ${1, 2, \ldots, n}$, construct a permutation $\pi$ of the stated form such that $\rho\pi$...
TAOCP 1.3.3 Exercise 29
Section 1.3.3: Applications to Permutations Exercise 29. [ M25 ] Prove that the cycle form of the Josephus permutation when $m = 2$ can be obtained by first expressing the "perfect shuffle" permutation of ${1, 2, \ldots, 2n}$, which takes $(1, 2, \ldots, 2n)$ into $(2, 4, \ldots, 2n, 1, 3, \ldots, 2n - 1)$, in cycle form, then reversing left and right and erasing all the numbers greater than...
TAOCP 1.3.3 Exercise 27
Section 1.3.3: Applications to Permutations Exercise 27. [ M20 ] Use the principle of inclusion and exclusion to count the number of integers $n$ in the range $0 \le n < am_1m_2\cdots m_t$ that are not divisible by any of $m_1, m_2, \ldots, m_t$. Here $m_1, m_2, \ldots, m_t$, and $a$ are positive integers, with $m_j \perp m_k$ when $j \ne k$. Verified: no Solve time: 5m39s Solution Let $M...
TAOCP 1.3.3 Exercise 26
Section 1.3.3: Applications to Permutations Exercise 26. [ M24 ] Extend the principle of inclusion and exclusion to obtain a formula for the number of elements that are in exactly $r$ of the subsets $S_1, S_2, \ldots, S_M$. (The text considers only the case $r = 0$.) Verified: no Solve time: 13m33s Solution For each element $x$ in the underlying universe, let $t(x)$ denote the number of sets among $S_1,S_2,\ldots,S_M$...
TAOCP 1.3.3 Exercise 21
Section 1.3.3: Applications to Permutations Exercise 21. [ M22 ] What is the probability $P(n;\alpha_1,\alpha_2,\ldots)$ that a permutation of $n$ objects has exactly $\alpha_1$ one-cycles, $\alpha_2$ two-cycles, etc.? Verified: no Solve time: 5m40s Solution Let $$ \alpha_1,\alpha_2,\ldots $$ denote the numbers of cycles of lengths $$ 1,2,\ldots, $$ respectively. A permutation of $n$ objects can have this cycle structure only if $$ \sum_{m\ge1} m\alpha_m=n. \tag{1} $$ We first count the...
TAOCP 1.3.3 Exercise 17
Section 1.3.3: Applications to Permutations Exercise 17. [ M24 ] (a) The text demonstrates that there are $n!H_n$ cycles altogether, among all the permutations on $n$ elements. If these cycles (including singleton cycles) are individually written on $n!H_n$ slips of paper, and if one of these slips of paper is chosen at random, what is the average length of the cycle that is thereby picked? (b) If we write the...
TAOCP 1.2.9 Exercise 22
Section 1.2.9: Generating Functions Exercise 22. [ M21 ] Find a generating function $G(z)$ for which $$ [z^n]G(z) = \sum_{k_0+2k_1+4k_2+8k_3+\cdots=n} \binom{r}{k_0}\binom{r}{k_1}\binom{r}{k_2}\binom{r}{k_3}\cdots . $$ Verified: yes Solve time: 1m04s Solution For each integer $j \ge 0$, define $$ A_j(z) = \sum_{k \ge 0} \binom{r}{k} z^{k 2^j}. $$ Each $A_j(z)$ is a power series in which the term corresponding to $k$ contributes exponent $k 2^j$ and coefficient $\binom{r}{k}$. Consider the infinite product...
TAOCP 1.2.9 Exercise 24
Section 1.2.9: Generating Functions Exercise 24. [ M22 ] Prove that, if $G(z)$ is any generating function, we have $$ \sum_k \binom{m}{k}[z^{,n-k}]G(z)^k = z^n ^m. $$ Evaluate both sides of this identity when $G(z)$ is (a) $1/(1-z)$; (b) $(e^z-1)/z$. Verified: yes Solve time: 1m11s Solution Expand the right-hand side using the binomial theorem and the definition of generating functions: $$ (1+zG(z))^m=\sum_{k=0}^m \binom{m}{k}(zG(z))^k =\sum_{k=0}^m \binom{m}{k} z^k G(z)^k. $$ For a fixed...
TAOCP 1.2.9 Exercise 25
Section 1.2.9: Generating Functions Exercise 25. [ M23 ] Evaluate the sum $$ \sum_k \binom{n}{k} 2^{n-2k}(-2)^k $$ by simplifying the equivalent formula $$ \sum_k w^k ^n z^{n-k} ^{2n-2k}. $$ Verified: no Solve time: 3m34s Solution Consider $$ \sum_k \binom{n}{k} 2^{,n-2k}(-2)^k. $$ Rewrite the factor depending on $k$: $$ 2^{,n-2k}(-2)^k = 2^n \cdot 2^{-2k} \cdot (-2)^k = 2^n \cdot \frac{(-2)^k}{4^k} = 2^n \cdot \left(-\frac{1}{2}\right)^k. $$ Substituting into the sum gives $$...
TAOCP 1.2.9 Exercise 20
Section 1.2.9: Generating Functions Exercise 20. [ M21 ] For what coefficients $c_{mk}$ is $$ \sum_{n \ge 0} n^m z^n = \sum_{k=0}^{m} c_{mk}\frac{z^k}{(1-z)^{k+1}} , ? $$ Verified: yes Solve time: 1m20s Solution Let $$ A_m(z)=\sum_{n\ge 0} n^m z^n. $$ For $m=0$, $$ A_0(z)=\sum_{n\ge 0} z^n=\frac{1}{1-z}. $$ For $m\ge 0$, use $$ z\frac{d}{dz} z^n = n z^n, $$ so repeated application gives $$ A_{m+1}(z)=z\frac{d}{dz}A_m(z). $$ Define coefficients $c_{mk}$ by $$ A_m(z)=\sum_{k=0}^m...
TAOCP 1.2.9 Exercise 17
Section 1.2.9: Generating Functions Exercise 17. [ M25 ] What are the coefficients of $1/(1-z)^w$ if this function is expanded into a double power series in terms of both $z$ and $w$? Verified: yes Solve time: 1m03s Solution Start from the identity $$ \frac{1}{(1-z)^w} = (1-z)^{-w} = \exp!\bigl(-w \ln(1-z)\bigr). $$ Rewrite the logarithm in the form $$ -\ln(1-z) = \ln\frac{1}{1-z}. $$ Hence $$ \frac{1}{(1-z)^w} = \exp!\left(w \ln\frac{1}{1-z}\right). $$ Expand the...
TAOCP 1.2.9 Exercise 16
Section 1.2.9: Generating Functions Exercise 16. [ M22 ] Give a simple formula for the generating function $$ G_n^{(r)}(z) = \sum_k a_{nkr} z^k, $$ where $a_{nkr}$ is the number of ways to choose $k$ out of $n$ objects, subject to the condition that each object may be chosen at most $r$ times. (If $r=1$, we have $\binom{n}{k}$ ways, and if $r \ge k$, we have the number of combinations with...
TAOCP 1.2.9 Exercise 18
Section 1.2.9: Generating Functions Exercise 18. [ M25 ] Given positive integers $n$ and $r$, find a simple formula for the value of the following sums: (a) $\sum_{1 \le k_1 < k_2 < \cdots < k_r \le n} k_1k_2\cdots k_r$; (b) $\sum_{1 \le k_1 \le k_2 \le \cdots \le k_r \le n} k_1k_2\cdots k_r$. (For example, when $n=3$ and $r=2$ the sums are, respectively, $1\cdot2 + 1\cdot3 + 2\cdot3$ and...
TAOCP 1.2.9 Exercise 10
Section 1.2.9: Generating Functions Exercise 10. [ M25 ] An elementary symmetric function is defined by the formula $$ e_m = \sum_{1 \le j_1 < \cdots < j_m \le n} x_{j_1}\cdots x_{j_m}. $$ (This is the same as $h_m$ of Eq. (33), except that equal subscripts are not allowed.) Find the generating function for $e_m$, and express $e_m$ in terms of the $S_j$ in Eq. (34). Write out the formulas...
TAOCP 1.2.9 Exercise 12
Section 1.2.9: Generating Functions Exercise 12. [ M20 ] Suppose we have a doubly subscripted sequence $\langle a_{mn} \rangle$ for $m,n=0,1,\ldots$; show how this double sequence can be represented by a single generating function of two variables, and determine the generating function for $\left\langle \binom{n}{m} \right\rangle$. Verified: yes Solve time: 1m24s Solution Let $\langle a_{mn} \rangle$ be a doubly indexed sequence for $m,n \ge 0$. Introduce two independent parameters $x$...
TAOCP 1.2.9 Exercise 11
Section 1.2.9: Generating Functions Exercise 11. [ M25 ] Equation (39) can also be used to express the $S$'s in terms of the $h$'s: We find $S_1=h_1$, $S_2=2h_2-h_1^2$, $S_3=3h_3-3h_1h_2+h_1^3$, etc. What is the coefficient of $h_1^{k_1}h_2^{k_2}\cdots h_m^{k_m}$ in this representation of $S_m$, when $k_1+2k_2+\cdots+mk_m=m$? Verified: yes Solve time: 1m46s Solution Let $$ H(z)=\sum_{m\ge 0} h_m z^m,\qquad h_0=1, $$ and let $$ S_j=\sum_i x_i^j. $$ From Eq. (39) in the text,...
TAOCP 1.2.9 Exercise 8
Section 1.2.9: Generating Functions Exercise 8. [ M23 ] Find the generating function for $p(n)$, the number of partitions of $n$. Verified: yes Solve time: 56s Solution Let $p(n)$ denote the number of representations of $n$ as a sum of positive integers, where order is disregarded and repetition is allowed. Each partition of $n$ is uniquely determined by a sequence $\langle a_1,a_2,a_3,\ldots\rangle$ of nonnegative integers such that $$ \sum_{k \ge...
TAOCP 1.2.9 Exercise 5
Section 1.2.9: Generating Functions Exercise 5. [ M20 ] Prove Eq. (23) by induction on $n$. Verified: yes Solve time: 1m26s Solution Define $S(k,n)$ as in Eq. (23), the Stirling numbers of the second kind. The claim is $$ (e^z-1)^n = n!\sum_{k \ge 0} \left{ {k \atop n} \right}\frac{z^k}{k!}. $$ Induction is carried out on $n$. For $n=0$, $(e^z-1)^0=1$. In the right-hand expression, $$ 0!\sum_{k \ge 0} \left{ {k \atop...
TAOCP 1.2.8 Exercise 40
Section 1.2.8: Fibonacci Numbers Exercise 40. [ M25 ] Solve the recurrence $$ f(1)=0; \qquad f(n)=\min_{0<k<n}\max(1+f(k),,2+f(n-k)), \qquad n>1. $$ Verified: no Solve time: 1m21s Solution Define $L(n)=\left\lceil \log_2 n \right\rceil + \left\lceil \log_2 (n+1) \right\rceil - 2$ for $n\ge 1$. Lemma 1 For every integer $n\ge 1$ and every integer $k$ with $0<k<n$, $$ \max(1+L(k),,2+L(n-k)) \ge L(n). $$ Let $a=\left\lceil \log_2 k \right\rceil$, $b=\left\lceil \log_2 (k+1) \right\rceil$, $c=\left\lceil \log_2 (n-k)...
TAOCP 1.2.8 Exercise 41
Section 1.2.8: Fibonacci Numbers Exercise 41. [ M25 ] (Yuri Matiyasevich, 1990.) Let $f(x)=\lfloor x+\phi^{-1}\rfloor$. Prove that if $$ n = F_{k_1} + \cdots + F_{k_r} $$ is the representation of $n$ in the Fibonacci number system of exercise 34, then $$ F_{k_1+1} + \cdots + F_{k_r+1} = f(\phi n). $$ Find a similar formula for $F_{k_1-1} + \cdots + F_{k_r-1}$. Verified: yes Solve time: 3m20s We start from a...
TAOCP 1.2.8 Exercise 39
Section 1.2.8: Fibonacci Numbers Exercise 39. [ M24 ] Find a closed form expression for $a_n$, given that $a_0=0$, $a_1=1$, and $a_{n+2}=a_{n+1}+6a_n$ for $n \ge 0$. Verified: yes Solve time: 1m12s Solution The recurrence $$ a_{n+2}=a_{n+1}+6a_n,\quad a_0=0,\quad a_1=1 $$ is linear with constant coefficients, so we seek solutions of the form $a_n=r^n$. Substitution into the recurrence gives $$ r^{n+2}=r^{n+1}+6r^n. $$ Dividing by $r^n$ (with $r\neq 0$) yields the characteristic equation...
TAOCP 1.2.8 Exercise 35
Section 1.2.8: Fibonacci Numbers Exercise 35. [ M24 ] ( A phi number system. ) Consider real numbers written with the digits $0$ and $1$ using base $\phi$; thus $(100.1)_\phi = \phi^2 + \phi^{-1}$. Show that there are infinitely many ways to represent the number $1$; for example, $$ 1 = (.11) \phi = (.011111\ldots) \phi. $$ But if we require that no two adjacent 1s occur and that the...
TAOCP 1.2.8 Exercise 32
Section 1.2.8: Fibonacci Numbers Exercise 32. [ M24 ] The remainder of one Fibonacci number divided by another is $\pm$ a Fibonacci number: Show that, modulo $F_n$, $$ F_{mn+r} \equiv \begin{cases} F_r, & \text{if } m \bmod 4 = 0; \ (-1)^{r+1}F_{n-r}, & \text{if } m \bmod 4 = 1; \ (-1)^nF_r, & \text{if } m \bmod 4 = 2; \ (-1)^{r+1+n}F_{n-r}, & \text{if } m \bmod 4 = 3....
TAOCP 1.2.8 Exercise 34
Section 1.2.8: Fibonacci Numbers Exercise 34. [ M24 ] ( The Fibonacci number system. ) Let the notation $k \gg m$ mean that $k \ge m+2$. Show that every positive integer $n$ has a unique representation $$ n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}, $$ where $k_1 \gg k_2 \gg \cdots \gg k_r \gg 0$. Verified: yes Solve time: 3m09s Assume the Fibonacci numbers are defined by $$...
TAOCP 1.2.8 Exercise 31
Section 1.2.8: Fibonacci Numbers Exercise 31. [ M20 ] Show that $F_{2n}\phi \bmod 1 = 1-\phi^{-2n}$ and $F_{2n+1}\phi \bmod 1 = \phi^{-2n-1}$. Verified: yes Solve time: 4m30s Start from Binet’s formula in a consistent and fully controlled way. Let $$ \phi = \frac{1+\sqrt{5}}{2}, \quad \psi = \frac{1-\sqrt{5}}{2} = -\phi^{-1}. $$ Then $$ F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}, \quad \phi + \psi = 1, \quad \phi\psi = -1, \quad \psi =...
TAOCP 1.2.8 Exercise 28
Section 1.2.8: Fibonacci Numbers Exercise 28. [ M21 ] What is $F_{n+1}-\phi F_n$? Verified: yes Solve time: 1m22s Solution Let $\hat{\phi} = \frac{1}{2}(1-\sqrt{5})$. From equation (14), $$ F_n = \frac{1}{\sqrt{5}}(\phi^n - \hat{\phi}^n), \qquad F_{n+1} = \frac{1}{\sqrt{5}}(\phi^{n+1} - \hat{\phi}^{n+1}). $$ Compute the difference: $$ F_{n+1} - \phi F_n = \frac{1}{\sqrt{5}}(\phi^{n+1} - \hat{\phi}^{n+1}) \phi \cdot \frac{1}{\sqrt{5}}(\phi^n - \hat{\phi}^n). $$ Combine terms over a common denominator: $$ F_{n+1} - \phi F_n = \frac{1}{\sqrt{5}}\left(\phi^{n+1}...
TAOCP 1.2.8 Exercise 29
Section 1.2.8: Fibonacci Numbers Exercise 29. [ M23 ] ( Fibonomial coefficients. ) Edouard Lucas defined the quantities $$ \binom{n}{k} {\mathcal{F}} = \frac{F_nF {n-1}\cdots F_{n-k+1}}{F_kF_{k-1}\cdots F_1} = \prod_{j=1}^{k}\left(\frac{F_{n-k+j}}{F_j}\right) $$ in a manner analogous to binomial coefficients. (a) Make a table of $\binom{n}{k} {\mathcal{F}}$ for $0 \le k \le n \le 6$. (b) Show that $\binom{n}{k} {\mathcal{F}}$ is always an integer because we have $$ \binom{n}{k} {\mathcal{F}} = F {k-1}\binom{n-1}{k}_{\mathcal{F}} F_{n-k+1}\binom{n-1}{k-1}_{\mathcal{F}}....
TAOCP 1.2.8 Exercise 25
Section 1.2.8: Fibonacci Numbers Exercise 25. [ M21 ] Show that $$ 2^nF_n = 2 \sum_{k\ \mathrm{odd}} \binom{n}{k} 5^{(k-1)/2}. $$ Verified: yes Solve time: 1m14s Solution From Eq. (14), the closed form for Fibonacci numbers is $$ F_n = \frac{1}{\sqrt{5}}(\phi^n - \hat{\phi}^n), $$ where $\phi = \frac{1}{2}(1+\sqrt{5})$ and $\hat{\phi} = \frac{1}{2}(1-\sqrt{5})$. Multiplying both sides by $2^n$ gives $$ 2^n F_n = \frac{1}{\sqrt{5}}\left((2\phi)^n - (2\hat{\phi})^n\right). $$ Using $2\phi = 1+\sqrt{5}$ and...
TAOCP 1.2.8 Exercise 26
Section 1.2.8: Fibonacci Numbers Exercise 26. [ M20 ] Using the previous exercise, show that $F_p \equiv 5^{(p-1)/2}$ (modulo $p$) if $p$ is an odd prime. Verified: yes Solve time: 1m43s Solution From Exercise 25 with $n=p$, where $p$ is an odd prime, we have $$ 2^p F_p = 2 \sum_{k\ \mathrm{odd}} \binom{p}{k} 5^{(k-1)/2}. $$ For a prime $p$, the binomial coefficients satisfy $$ \binom{p}{k} \equiv 0 \pmod p \quad...
TAOCP 1.2.8 Exercise 27
Section 1.2.8: Fibonacci Numbers Exercise 27. [ M20 ] Using the previous exercise, show that if $p$ is a prime different from $5$, then either $F_{p-1}$ or $F_{p+1}$ (not both) is a multiple of $p$. Verified: yes Solve time: 1m28s Solution From Exercise 26, for any odd prime $p \ne 5$, $$ F_p \equiv 5^{(p-1)/2} \pmod p. $$ By Euler’s criterion, $$ 5^{(p-1)/2} \equiv \left(\frac{5}{p}\right) \pmod p, $$ and the...
TAOCP 1.2.8 Exercise 23
Section 1.2.8: Fibonacci Numbers Exercise 23. [ M23 ] Generalizing the preceding exercise, show that $$ \sum_k \binom{n}{k} F_t^k F_{t-1}^{,n-k} F_{m+k} $$ is always a Fibonacci number. Verified: yes Solve time: 3m24s We restart from the definition and avoid splitting into $F_k$ and $F_{k-1}$, which is the source of the earlier error. Let $$ S=\sum_{k=0}^n \binom{n}{k} F_t^k F_{t-1}^{,n-k} F_{m+k}. $$ We use Binet’s formula: $$ F_r=\frac{1}{\sqrt{5}}(\phi^r-\hat{\phi}^r), \quad \phi=\frac{1+\sqrt{5}}{2},\quad \hat{\phi}=\frac{1-\sqrt{5}}{2}. $$...
TAOCP 1.2.8 Exercise 21
Section 1.2.8: Fibonacci Numbers Exercise 21. [ M25 ] What is $\sum_{k=0}^{n} F_kx^k$? Verified: yes Solve time: 3m08s Let $$ S_n(x)=\sum_{k=0}^{n}F_k x^k. $$ We determine $S_n(x)$ directly from the Fibonacci recurrence $$ F_k = F_{k-1}+F_{k-2}\quad (k\ge 2), $$ with $F_0=0,;F_1=1$. Step 1: Multiply by $1-x-x^2$ Compute $$ (1-x-x^2)S_n(x)=S_n(x)-xS_n(x)-x^2S_n(x). $$ Write each term explicitly. $$ S_n(x)=\sum_{k=0}^{n}F_k x^k. $$ $$ xS_n(x)=\sum_{k=0}^{n}F_k x^{k+1}=\sum_{k=1}^{n+1}F_{k-1}x^k. $$ $$ x^2S_n(x)=\sum_{k=0}^{n}F_k x^{k+2}=\sum_{k=2}^{n+2}F_{k-2}x^k. $$ Hence $$ (1-x-x^2)S_n(x) \sum_{k=0}^{n}F_k x^k...
TAOCP 1.2.8 Exercise 22
Section 1.2.8: Fibonacci Numbers Exercise 22. [ M20 ] Show that $\sum_k \binom{n}{k}F_{m+k}$ is a Fibonacci number. Verified: yes Solve time: 3m27s Let $$ M=\begin{pmatrix}1&1\[2pt]1&0\end{pmatrix}. $$ A standard identity for Fibonacci numbers is $$ M^k= \begin{pmatrix} F_{k+1} & F_k\ F_k & F_{k-1} \end{pmatrix} \qquad (k\ge 1), $$ with the convention $F_0=0, F_1=1$. This follows by induction from the recurrence $F_{k+1}=F_k+F_{k-1}$. Hence, $$ F_{m+k} = (M^{m+k})_{1,2}. $$ Step 1: Rewrite the...
TAOCP 1.2.8 Exercise 20
Section 1.2.8: Fibonacci Numbers Exercise 20. [ M16 ] Express $\sum_{k=0}^{n} F_k$ in terms of Fibonacci numbers. Verified: yes Solve time: 1m Solution Let $$ S_n = \sum_{k=0}^{n} F_k. $$ Define $$ T_n = F_{n+2} - 1. $$ The Fibonacci recurrence $F_{n+2} = F_{n+1} + F_n$ gives $$ T_n = F_{n+1} + F_n - 1 = (F_{n+1} - 1) + F_n, $$ so $$ T_n - T_{n-1} = F_n \quad...
TAOCP 1.2.8 Exercise 17
Section 1.2.8: Fibonacci Numbers Exercise 17. [ M24 ] Using the conventions of exercise 8, prove the following generalization of Eq. (4): $$ F_{n+k}F_{m-k} - F_nF_m = (-1)^n F_{m-n-k}F_k. $$ Verified: yes Solve time: 2m Solution From identity (6) in Section 1.2.8, $$ F_{r+s} = F_s F_{r+1} + F_{s-1}F_r, $$ we obtain expansions of shifted Fibonacci numbers in terms of lower indices. First expand $F_{n+k}$ by taking $r=n$ and $s=k$:...
TAOCP 1.2.8 Exercise 13
Section 1.2.8: Fibonacci Numbers Exercise 13. [ M22 ] Express the following sequences in terms of the Fibonacci numbers, when $r$, $s$, and $c$ are given constants: a) $a_0=r$, $a_1=s$; $a_{n+2}=a_{n+1}+a_n$, for $n \ge 0$. b) $b_0=0$, $b_1=1$; $b_{n+2}=b_{n+1}+b_n+c$, for $n \ge 0$. Verified: yes Solve time: 1m37s Solution (a) Let $a_0=r$, $a_1=s$, and $a_{n+2}=a_{n+1}+a_n$ for $n\ge 0$. The Fibonacci numbers satisfy $F_0=0$, $F_1=1$, and $F_{n+2}=F_{n+1}+F_n$. Consider the expression $$...
TAOCP 1.2.8 Exercise 16
Section 1.2.8: Fibonacci Numbers Exercise 16. [ M20 ] Fibonacci numbers appear implicitly in Pascal's triangle if it is viewed from the right angle. Show that the following sum of binomial coefficients is a Fibonacci number: $$ \sum_{k=0}^{n} \binom{n-k}{k}. $$ Verified: yes Solve time: 1m21s Solution Let $$ S_n=\sum_{k=0}^{n}\binom{n-k}{k}. $$ The summand vanishes when $k>n-k$, since then $\binom{n-k}{k}=0$. Hence the sum is finite and well defined. We shall show that...
TAOCP 1.2.8 Exercise 15
Section 1.2.8: Fibonacci Numbers Exercise 15. [ M22 ] Let $f(n)$ and $g(n)$ be arbitrary functions, and for $n \ge 0$ let $$ \begin{aligned} a_0&=0, & a_1&=1, & a_{n+2}&=a_{n+1}+a_n+f(n);\ b_0&=0, & b_1&=1, & b_{n+2}&=b_{n+1}+b_n+g(n);\ c_0&=0, & c_1&=1, & c_{n+2}&=c_{n+1}+c_n+xf(n)+yg(n). \end{aligned} $$ Express $c_n$ in terms of $x$, $y$, $a_n$, $b_n$, and $F_n$. Verified: yes Solve time: 1m24s Solution Let $A_n$ and $B_n$ be defined by $$ A_n = a_n -...
TAOCP 1.2.8 Exercise 9
Section 1.2.8: Fibonacci Numbers Exercise 9. [ M20 ] Using the conventions of exercise 8, determine whether Eqs. (4), (6), (14), and (15) still hold when the subscripts are allowed to be any integers. Verified: yes Solve time: 1m49s Solution The Fibonacci sequence is extended to all integers $n$ by the recurrence $$ F_{n+2}=F_{n+1}+F_n $$ for all integers $n$, together with the initial values $F_0=0$ and $F_1=1$. From the recurrence...
TAOCP 1.2.8 Exercise 11
Section 1.2.8: Fibonacci Numbers Exercise 11. [ M20 ] Show that $\phi^n = F_n\phi + F_{n-1}$ and $\hat{\phi}^n = F_n\hat{\phi} + F_{n-1}$, for all integers $n$. Verified: yes Solve time: 2m46s We prove both identities by a direct induction using the Fibonacci recurrence, avoiding any manipulation that mixes the two sides prematurely. Recall the Fibonacci numbers satisfy $$ F_{n+1} = F_n + F_{n-1} $$ for all integers $n$, together with...
TAOCP 1.2.7 Exercise 25
Section 1.2.7: Harmonic Numbers Exercise 25. [ M21 ] Let $$ H_n^{(u,v)} = \sum_{1 \le j \le k \le n} \frac{1}{j^u k^v}. $$ What are $H_n^{(0,v)}$ and $H_n^{(u,0)}$? Prove the general identity $$ H_n^{(u,v)} + H_n^{(v,u)} = H_n^{(u)}H_n^{(v)} + H_n^{(u+v)}. $$ Verified: yes Solve time: 1m33s Solution Let $H_n^{(u)}=\sum_{k=1}^{n}\frac{1}{k^u}$. Then $$ H_n^{(u,v)}=\sum_{1\le j\le k\le n}\frac{1}{j^u k^v}. $$ Evaluation of $H_n^{(0,v)}$ Since $j^0=1$, $$ H_n^{(0,v)}=\sum_{1\le j\le k\le n}\frac{1}{k^v}. $$ For fixed...
TAOCP 1.2.7 Exercise 21
Section 1.2.7: Harmonic Numbers Exercise 21. [ M24 ] Evaluate $\sum_{k=1}^{n} H_k/(n+1-k)$. Verified: no Solve time: 1m10s Solution Let $$ S=\sum_{k=1}^{n}\frac{H_k}{n+1-k}. $$ Set $j=n+1-k$, so $k=n+1-j$ and $j$ runs from $1$ to $n$. Then $$ S=\sum_{j=1}^{n}\frac{H_{n+1-j}}{j}. $$ For each $j$, $$ H_n = H_{n+1-j} + \sum_{t=n+2-j}^{n}\frac{1}{t}, $$ hence $$ H_{n+1-j}=H_n-\sum_{t=n+2-j}^{n}\frac{1}{t}. $$ Substitution yields $$ S=\sum_{j=1}^{n}\frac{H_n}{j}-\sum_{j=1}^{n}\frac{1}{j}\sum_{t=n+2-j}^{n}\frac{1}{t}. $$ The first term equals $H_n^2$. For the double sum, interchange order: for fixed $t$,...
TAOCP 1.2.7 Exercise 17
Section 1.2.7: Harmonic Numbers Exercise 17. [ M24 ] (E. Waring, 1782.) Let $p$ be an odd prime. Show that the numerator of $H_{p-1}$ is divisible by $p$. Verified: no Solve time: 2m39s Solution Let $p$ be an odd prime and consider $$ H_{p-1}=\sum_{k=1}^{p-1}\frac{1}{k}. $$ Split the sum into pairs of indices $k$ and $p-k$ for $k=1,2,\dots,\frac{p-1}{2}$. This gives $$ H_{p-1}=\sum_{k=1}^{(p-1)/2}\left(\frac{1}{k}+\frac{1}{p-k}\right). $$ For each $k$ in this range, $$ \frac{1}{k}+\frac{1}{p-k}...
TAOCP 1.2.7 Exercise 13
Section 1.2.7: Harmonic Numbers Exercise 13. [ M22 ] Prove the identity $$ \sum_{k=1}^{n}\frac{x^k}{k} = H_n + \sum_{k=1}^{n}\binom{n}{k}\frac{(x-1)^k}{k}. $$ Verified: yes Solve time: 1m05s Solution Start with the binomial expansion $$ x^k = (1+(x-1))^k = \sum_{j=0}^{k}\binom{k}{j}(x-1)^j. $$ Divide by $k$ and sum over $1 \le k \le n$: $$ \sum_{k=1}^{n}\frac{x^k}{k} = \sum_{k=1}^{n}\frac{1}{k}\sum_{j=0}^{k}\binom{k}{j}(x-1)^j. $$ Interchanging the order of summation yields $$ \sum_{k=1}^{n}\frac{x^k}{k} = \sum_{k=1}^{n}\frac{1}{k} \sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{k}\binom{k}{j}(x-1)^j. $$ The first term equals $H_n$...
TAOCP 1.2.7 Exercise 14
Section 1.2.7: Harmonic Numbers Exercise 14. [ M22 ] Show that $$ \sum_{k=1}^{n}\frac{H_k}{k} = \frac{1}{2}(H_n^2 + H_n^{(2)}), $$ and evaluate $\sum_{k=1}^{n} H_k/(k+1)$. Verified: yes Solve time: 1m38s Solution Let $$ S=\sum_{k=1}^{n}\frac{H_k}{k}. $$ Using $H_k=\sum_{j=1}^{k}\frac{1}{j}$, $$ S=\sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{jk}. $$ The index set is $1\le j\le k\le n$, so $$ S=\sum_{1\le j\le k\le n}\frac{1}{jk}. $$ Split the full square sum $$ \sum_{j=1}^{n}\sum_{k=1}^{n}\frac{1}{jk} $$ into regions $j\le k$ and $j>k$. The full sum equals...
TAOCP 1.2.7 Exercise 15
Section 1.2.7: Harmonic Numbers Exercise 15. [ M23 ] Express $\sum_{k=1}^{n} H_k^2$ in terms of $n$ and $H_n$. Verified: yes Solve time: 1m21s Solution Let $$ S_n = \sum_{k=1}^{n} H_k^2. $$ Expand $H_k$ as a partial sum, $$ H_k = \sum_{i=1}^{k}\frac{1}{i}, \quad H_k^2 = \sum_{i=1}^{k}\sum_{j=1}^{k}\frac{1}{ij}. $$ Hence $$ S_n = \sum_{k=1}^{n}\sum_{i=1}^{k}\sum_{j=1}^{k}\frac{1}{ij}. $$ Interchanging the order of summation over $k$ yields $$ S_n = \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=\max(i,j)}^{n}\frac{1}{ij} = \sum_{i=1}^{n}\sum_{j=1}^{n}\frac{n-\max(i,j)+1}{ij}. $$ Split the double...
TAOCP 1.2.7 Exercise 11
Section 1.2.7: Harmonic Numbers Exercise 11. [ M21 ] Using summation by parts, evaluate $$ \sum_{1<k \le n}\frac{1}{k(k-1)}H_k. $$ Verified: yes Solve time: 1m37s Solution Let $$ S=\sum_{1<k\le n}\frac{1}{k(k-1)}H_k =\sum_{k=2}^{n}\frac{1}{k(k-1)}H_k. $$ Set $j=k-1$. Then $k=j+1$ and $j$ runs from $1$ to $n-1$. Hence $$ S=\sum_{j=1}^{n-1}\frac{1}{j(j+1)}H_{j+1}. $$ Write $$ \frac{1}{j(j+1)}=\frac{1}{j}-\frac{1}{j+1}. $$ Define $a_j=\frac{1}{j}$ and $b_j=H_{j+1}$. Then $$ S=\sum_{1\le j<n}(a_j-a_{j+1})b_j. $$ Apply the summation by parts identity from Exercise 1.2.7.10: $$ \sum_{1...
TAOCP 1.2.7 Exercise 9
Section 1.2.7: Harmonic Numbers Exercise 9. [ M18 ] Theorem A applies only when $x>0$; what is the value of the sum considered when $x=-1$? Verified: yes Solve time: 1m15s Solution Let $$ S_n=\sum_{k=1}^{n}\binom{n}{k}(-1)^k H_k. $$ Using $H_k=\sum_{j=1}^{k}\frac{1}{j}$, $$ S_n=\sum_{k=1}^{n}\binom{n}{k}(-1)^k \sum_{j=1}^{k}\frac{1}{j} =\sum_{j=1}^{n}\frac{1}{j}\sum_{k=j}^{n}(-1)^k\binom{n}{k}. $$ The inner sum is evaluated by splitting the full binomial sum: $$ \sum_{k=0}^{n}(-1)^k\binom{n}{k}=(1-1)^n=0, $$ so $$ \sum_{k=j}^{n}(-1)^k\binom{n}{k}=-\sum_{k=0}^{j-1}(-1)^k\binom{n}{k}. $$ The partial alternating binomial identity $$ \sum_{k=0}^{m}(-1)^k\binom{n}{k}=(-1)^m\binom{n-1}{m} $$ applied...
TAOCP 1.2.7 Exercise 10
Section 1.2.7: Harmonic Numbers Exercise 10. [ M20 ] (Summation by parts.) Prove the general formula $$ \sum_{1 \le k < n}(a_{k+1}-a_k)b_k = a_nb_n - a_1b_1 - \sum_{1 \le k < n}a_{k+1}(b_{k+1}-b_k). $$ Verified: yes Solve time: 1m28s Solution Let $$ S=\sum_{1\le k<n}(a_{k+1}-a_k)b_k. $$ Expanding the product yields $$ S=\sum_{1\le k<n}a_{k+1}b_k-\sum_{1\le k<n}a_k b_k. $$ In the first sum, substitute $j=k+1$, giving $$ \sum_{1\le k<n}a_{k+1}b_k=\sum_{2\le j\le n}a_j b_{j-1}. $$ Hence $$...
TAOCP 1.2.7 Exercise 7
Section 1.2.7: Harmonic Numbers Exercise 7. [ M21 ] Let $T(m,n)=H_m+H_n-H_{mn}$. Show that when $m$ or $n$ increases, $T(m,n)$ never increases (assuming that $m$ and $n$ are positive). Compute the minimum and maximum values of $T(m,n)$ for $m,n>0$. Verified: yes Solve time: 2m46s Define $$ T(m,n)=H_m+H_n-H_{mn}, \qquad m,n\in \mathbb{Z}_{>0}. $$ 1. Monotonicity in each variable Fix $n$. Using $H_{m+1}=H_m+\frac{1}{m+1}$, $$ T(m+1,n)-T(m,n)=\frac{1}{m+1}-\bigl(H_{(m+1)n}-H_{mn}\bigr). $$ Expand the harmonic difference: $$ H_{(m+1)n}-H_{mn}=\sum_{k=mn+1}^{(m+1)n}\frac{1}{k}. $$ For...
TAOCP 1.2.7 Exercise 3
Section 1.2.7: Harmonic Numbers Exercise 3. [ M21 ] Generalize the argument used in the previous exercise to show that, for $r>1$, the sum $H_n^{(r)}$ remains bounded for all $n$. Find an upper bound. Verified: yes Solve time: 1m09s Solution Let $$ H_n^{(r)} = \sum_{k=1}^{n} \frac{1}{k^r}, \qquad r>1. $$ For each integer $j \ge 0$, consider the block of indices $$ 2^j \le k \le 2^{j+1}-1. $$ For every $k$...
TAOCP 1.2.6 Exercise 68
Section 1.2.6: Binomial Coefficients Exercise 68. [ M25 ] (A. de Moivre.) Prove that, if $n$ is a nonnegative integer, $$ \sum_k \binom{n}{k}p^k(1-p)^{n-k}|k-np| = 2\lfloor np \rfloor \binom{n}{\lfloor np \rfloor}p^{\lfloor np \rfloor}(1-p)^{n+1-\lfloor np \rfloor}. $$ Verified: no Solve time: 5m10s The earlier response failed because it replaced the task instead of proving the stated identity. The correct approach is to work directly from the given sum and show it reduces...
TAOCP 1.2.6 Exercise 67
Section 1.2.6: Binomial Coefficients Exercise 67. [ M20 ] Prove the upper bound $$ \binom{n}{k} \le \left(\frac{ne}{k}\right)^k, \qquad n \ge k \ge 0. $$ Verified: yes Solve time: 1m23s Solution For $k = 0$, the binomial coefficient satisfies $\binom{n}{0} = 1$. The right-hand expression is interpreted as $1$ since $\left(\frac{ne}{k}\right)^k$ equals $1$ at $k=0$ by the limiting convention $a^0 = 1$ for $a \ne 0$. Thus the inequality holds when...
TAOCP 1.2.6 Exercise 64
Section 1.2.6: Binomial Coefficients Exercise 64. [ M20 ] Show that $\left{{n \atop m}\right}$ is the number of ways to partition a set of $n$ elements into $m$ nonempty disjoint subsets. Verified: yes Solve time: 1m13s Solution Let $S(n,m)$ denote the number of ways to partition a set of $n$ elements into $m$ nonempty disjoint subsets. Let the underlying set be ${1,2,\dots,n}$. Fix $n \ge 1$ and $m \ge 1$,...
TAOCP 1.2.6 Exercise 62
Section 1.2.6: Binomial Coefficients Exercise 62. [ M23 ] Prove the identity $$ \sum_k (-1)^k \binom{l+m}{l+k}\binom{m+n}{m+k}\binom{n+l}{n+k} = \frac{(l+m+n)!}{l!,m!,n!}, \qquad \text{integer } l,m,n \ge 0. $$ Verified: no Solve time: 5m19s The previous argument failed because it repeatedly used coefficient manipulations that do not respect dependence on the summation index $k$, and it invoked an unproved alternating Vandermonde identity. A correct proof must keep the dependence on $k$ under control and...
TAOCP 1.2.6 Exercise 61
Section 1.2.6: Binomial Coefficients Exercise 61. [ M25 ] Evaluate the sum $$ \sum_k \left[{n+1 \atop k+1}\right]\left{{k \atop m}\right}(-1)^{k-m}, $$ thereby obtaining a companion formula for Eq. (55). Verified: yes Solve time: 5m14s 1. Where the previous solution fails The key failure is the incorrect “recognition step.” The quantity $$ S(n,m)=\sum_k \left[{n+1 \atop k+1}\right]\left{{k \atop m}\right}(-1)^{k-m} $$ is not another Stirling number of the second kind. The earlier argument incorrectly...
TAOCP 1.2.6 Exercise 58
Section 1.2.6: Binomial Coefficients Exercise 58. [ M23 ] In the notation of Eq. (40), prove the $q$-nomial theorem: $$ (1+x)(1+qx)\cdots(1+q^{n-1}x) = \sum_k \binom{n}{k}_q q^{k(k-1)/2}x^k. $$ Verified: yes Solve time: 1m10s Solution Let $$ P_n(x)=\prod_{j=0}^{n-1}(1+q^j x), \qquad n\ge 0. $$ Let $$ S_n(x)=\sum_{k\ge 0}\binom{n}{k}_q q^{k(k-1)/2}x^k. $$ The proof proceeds by showing that $P_n(x)=S_n(x)$ for all integers $n\ge 0$. For $n=0$, $$ P_0(x)=1, \qquad S_0(x)=\binom{0}{0}_q q^0 x^0=1, $$ since $\binom{0}{0}_q=1$ and...
TAOCP 1.2.6 Exercise 59
Section 1.2.6: Binomial Coefficients Exercise 59. [ M25 ] A sequence of numbers $A_{nk}$ satisfies $$ A_{n0}=1,\qquad A_{0k}=\delta_{0k},\qquad A_{nk}=A_{n-1,k}+A_{n-1,k-1}+\binom{n}{k}. $$ Find $A_{nk}$. Verified: yes Solve time: 4m34s The core issue is that the recurrence is valid only for $k \ge 1$. The boundary condition $A_{n0}=1$ is not propagated by the recurrence and must be enforced separately throughout the generating-function derivation. 1. Correct formulation of the generating function Define the full...
TAOCP 1.2.6 Exercise 60
Section 1.2.6: Binomial Coefficients Exercise 60. [ M23 ] How many $k$-combinations of $n$ objects are there, if repetition is allowed? Verified: yes Solve time: 1m37s Solution Let the $n$ objects be labeled $1,2,\ldots,n$. A $k$-combination with repetition is a selection of $k$ elements from this set where each object may be chosen multiple times, and order is irrelevant. Such a combination is completely determined by the multiplicities $$ x_1,x_2,\ldots,x_n,...
TAOCP 1.2.6 Exercise 57
Section 1.2.6: Binomial Coefficients Exercise 57. [ M22 ] Show that the coefficient $a_m$ in Stirling's attempt at generalizing the factorial function is $$ \frac{(-1)^m}{m!}\sum_{k>1}(-1)^k\binom{m-1}{k-1}\ln k. $$ Verified: no Solve time: 3m Let $f(n)=\ln(n!)$ and $\Delta f(n)=f(n+1)-f(n)$. For integers $m\ge 1$, $$ a_m=\frac{1}{m!}\Delta^m f(1). $$ 1. Forward difference expansion The $m$-th forward difference at $1$ is $$ \Delta^m f(1)=\sum_{k=0}^m (-1)^{m-k}\binom{m}{k}\ln((k+1)!). \tag{1} $$ Write $$ \ln((k+1)!)=\sum_{j=1}^{k+1}\ln j, $$ and substitute into...
TAOCP 1.2.6 Exercise 55
Section 1.2.6: Binomial Coefficients Exercise 55. [ M21 ] Considering each of Stirling's triangles as matrices, determine their inverses. Verified: yes Solve time: 58s Solution Let $S(n,k)$ denote Stirling numbers of the second kind and $s(n,k)$ denote Stirling numbers of the first kind in the signed form. Define two infinite lower-triangular matrices $S$ and $T$ by $$ S_{n,k} = S(n,k), \qquad T_{n,k} = s(n,k), \qquad n,k \ge 0, $$ with...
TAOCP 1.2.6 Exercise 54
Section 1.2.6: Binomial Coefficients Exercise 54. [ M21 ] Consider Pascal's triangle as a matrix. What is the inverse of that matrix? Verified: yes Solve time: 1m25s Solution Let $P = (p_{ij})_{i,j \ge 0}$ denote Pascal’s triangle matrix defined by $$ p_{ij} = \begin{cases} \binom{i}{j}, & i \ge j,\ 0, & i < j. \end{cases} $$ This is a lower triangular matrix with diagonal entries $p_{ii} = 1$. Let $Q...
TAOCP 1.2.6 Exercise 51
Section 1.2.6: Binomial Coefficients Exercise 51. [ M21 ] Prove Abel's formula, Eq. (16), by writing $y=(x+y)-x$, expanding the right-hand side in powers of $(x+y)$, and applying the result of the previous exercise. Verified: yes Solve time: 3m58s Let $z=x+y$. We prove Abel’s formula $$ (x+y)^{\overline r} \sum_{k=0}^r \binom{r}{k}, x^{\overline{r-k}}, y^{\overline{k}} $$ by showing that the right-hand side equals $z^{\overline r}$ as a polynomial in $z$. Define $$ F(z) \sum_{k=0}^r...
TAOCP 1.2.6 Exercise 53
Section 1.2.6: Binomial Coefficients Exercise 53. [ M25 ] Prove by induction on $m$ that $$ \sum_{k=0}^{m}\binom{r}{k}\binom{s}{n-k}\left(nr-(r+s)k\right) = (m+1)(n-m)\binom{r}{m+1}\binom{s}{n-m}. $$ Then use related formulas to derive $$ \sum_{k=0}^{m}\binom{2k-1}{k}\binom{2n-2k}{n-k}\frac{-1}{2k-1} = \frac{n-m}{2n}\binom{2m}{m}\binom{2n-2m}{n-m} \frac{1}{2}\binom{2n}{n}. $$ Verified: no Solve time: 5m24s Solution to Exercise 1.2.6.53 1. First identity (induction on $m$) Let $$ S_m=\sum_{k=0}^{m}\binom{r}{k}\binom{s}{n-k}\bigl(nr-(r+s)k\bigr). $$ We prove $$ S_m=(m+1)(n-m)\binom{r}{m+1}\binom{s}{n-m}. $$ Base case $m=0$ $$ S_0=\binom{r}{0}\binom{s}{n}(nr)=nr\binom{s}{n}. $$ Right-hand side: $$ (1)(n)\binom{r}{1}\binom{s}{n}=nr\binom{s}{n}. $$ So the base...
TAOCP 1.2.6 Exercise 48
Section 1.2.6: Binomial Coefficients Exercise 48. [ M25 ] Show that $$ \sum_{k \ge 0}\binom{n}{k}\frac{(-1)^k}{k+x} = \frac{n!}{x(x+1)\cdots(x+n)} = \frac{1}{x\binom{n+x}{n}}, $$ if the denominators are not zero. Verified: yes Solve time: 1m10s Solution Assume $x \notin {0,-1,-2,\ldots,-n}$ so that all denominators are nonzero. Start from the identity $$ \frac{1}{k+x}=\int_{0}^{1} t^{k+x-1},dt, \qquad x>0, $$ and note that both sides extend as meromorphic functions of $x$, so the resulting identity continues to hold...
TAOCP 1.2.6 Exercise 49
Section 1.2.6: Binomial Coefficients Exercise 49. [ M20 ] Show that the identity $(1+x)^r = (1-x^2)^r(1-x)^{-r}$ implies a relation on binomial coefficients. Verified: yes Solve time: 1m11s Solution From the binomial theorem (13), interpreted as formal power series in $x$, we have $$ (1+x)^r = \sum_{n \ge 0} \binom{r}{n} x^n. $$ The identity $$ (1+x)^r = (1-x^2)^r(1-x)^{-r} $$ is expanded by applying (13) to each factor. First, $$ (1-x^2)^r =...
TAOCP 1.2.6 Exercise 47
Section 1.2.6: Binomial Coefficients Exercise 47. [ M21 ] Given that $k$ is an integer, show that $$ \binom{r}{k}\binom{r-1/2}{k} = \binom{2r}{k}\binom{2r-k}{k}/4^k = \binom{2r}{2k}\binom{2k}{k}/4^k. $$ Give a simpler formula for the special case $r=-1/2$. Verified: yes Solve time: 2m43s Let $k \ge 0$ be an integer and $r$ an arbitrary parameter. 1. Main identity Start from the falling factorial definition $$ \binom{r}{k}=\frac{r^{\underline{k}}}{k!}, \qquad r^{\underline{k}}=r(r-1)\cdots(r-k+1). $$ Then $$ \binom{r}{k}\binom{r-\tfrac12}{k} =\frac{r^{\underline{k}},(r-\tfrac12)^{\underline{k}}}{(k!)^2} =\frac{1}{(k!)^2}\prod_{j=0}^{k-1}(r-j)\left(r-\tfrac12-j\right). $$...
TAOCP 1.2.6 Exercise 50
Section 1.2.6: Binomial Coefficients Exercise 50. [ M20 ] Prove Abel's formula, Eq. (16), in the special case $x+y=0$. Verified: no Solve time: 4m15s Abel’s formula (Eq. (16)) in TAOCP can be written in rising–factorial form as $$ \sum_{k=0}^n \binom{n}{k},(x) k,(y) {n-k} ;=; (x+y)_n, $$ where $$ (a)_m = a(a+1)\cdots(a+m-1), \qquad (a)_0 = 1. $$ Assume $x+y=0$, so $y=-x$. Substituting into the right-hand side gives $$ (x+y)_n = (0)_n. $$...
TAOCP 1.2.6 Exercise 46
Section 1.2.6: Binomial Coefficients Exercise 46. [ M21 ] Using Stirling's approximation, Eq. 1.2.5--(7), find an approximate value of $\binom{x+y}{y}$, assuming that both $x$ and $y$ are large. In particular, find the approximate size of $\binom{2n}{n}$ when $n$ is large. Verified: yes Solve time: 1m13s Solution Stirling’s approximation (Eq. 1.2.5–(7)) states $$ n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \qquad (n \to \infty). $$ Approximation of $\binom{x+y}{y}$ Starting from the factorial form (Eq....
TAOCP 1.2.6 Exercise 34
Section 1.2.6: Binomial Coefficients Exercise 34. [ M23 ] Show that Abel's generalization, Eq. (16), of the binomial formula is true also for rising powers: $$ (x+y)^{\overline{n}} = \sum_k \binom{n}{k}x(x-kz+1)^{\overline{k-1}}(y+kz)^{\overline{n-k}}. $$ Verified: no Solve time: 4m34s Let $$ R_n(x,y)=\sum_{k} \binom{n}{k},x,(x-kz+1)^{\overline{k-1}}(y+kz)^{\overline{n-k}}. $$ We prove $$ R_n(x,y)=(x+y)^{\overline{n}} $$ by induction on $n$. To avoid incorrect manipulations, we first remove the misleading algebra involving products of rising factorials. Step 1: Normalize the expression...
TAOCP 1.2.6 Exercise 35
Section 1.2.6: Binomial Coefficients Exercise 35. [ M23 ] Prove the addition formulas (46) for Stirling numbers directly from the definitions, Eqs. (44) and (45). Verified: yes Solve time: 1m22s Solution Let the Stirling numbers of the second kind $S(n,k)$ and the Stirling numbers of the first kind $s(n,k)$ be defined by Eqs. (44) and (45) of the text: $$ x^n=\sum_k S(n,k),x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_k s(n,k),x^k, $$ with summation over all...
TAOCP 1.2.6 Exercise 33
Section 1.2.6: Binomial Coefficients Exercise 33. [ M20 ] Show that the binomial formula is valid also when it involves factorial powers instead of the ordinary powers. In other words, prove that $$ (x+y)^{\underline{n}} = \sum_k \binom{n}{k}x^{\underline{k}}y^{\underline{n-k}}, \qquad (x+y)^{\overline{n}} = \sum_k \binom{n}{k}x^{\overline{k}}y^{\overline{n-k}}. $$ Verified: yes Solve time: 1m24s Solution Falling factorial identity Proceed by induction on $n$. For $n=0$, the identity reduces to $$ (x+y)^{\underline 0} = 1 $$ and...
TAOCP 1.2.6 Exercise 32
Section 1.2.6: Binomial Coefficients Exercise 32. [ M20 ] Show that $\sum_k \binom{n}{k}x^k = x^{\overline{n}}$, where $x^{\overline{n}}$ is the rising factorial power defined in Eq. 1.2.5--(19). Verified: yes Solve time: 2m40s The stated identity is incorrect, so no proof of it can be completed as written. The error is not a gap in reasoning but a false equation. 1. Disproof of the stated identity The exercise claims $$ \sum_k \binom{n}{k}...
TAOCP 1.2.6 Exercise 29
Section 1.2.6: Binomial Coefficients Exercise 29. [ M20 ] Show that Eq. (34) is just a special case of the general identity proved in exercise 1.2.3--33. Verified: yes Solve time: 1m18s Solution Let Eq. (34) denote the identity $$ \sum_k \binom{r}{k}\binom{s}{n-k} = \binom{r+s}{n}, $$ with the usual convention that terms with invalid binomial coefficients vanish. From the binomial theorem, $$ (1+x)^r = \sum_k \binom{r}{k} x^k, \qquad (1+x)^s = \sum_k \binom{s}{k}...
TAOCP 1.2.6 Exercise 30
Section 1.2.6: Binomial Coefficients Exercise 30. [ M24 ] Show that there is a better way to solve Example 3 than the way used in the text, by manipulating the sum so that Eq. (26) applies. Verified: yes Solve time: 2m42s Let Example 3 define $$ S=\sum_{k} \binom{r}{k}\binom{s}{m+k}, $$ where the sum is over all integers $k$ for which the binomial coefficients are defined, i.e., $0\le k\le r$ and $0\le...
TAOCP 1.2.6 Exercise 31
Section 1.2.6: Binomial Coefficients Exercise 31. [ M20 ] Evaluate $$ \sum_k \binom{m-r+s}{k}\binom{n+r-s}{n-k}\binom{r+k}{m+n} $$ in terms of $r$, $s$, $m$, and $n$, given that $m$ and $n$ are integers. Begin by replacing $$ \binom{r+k}{m+n} $$ by $\sum_j \binom{r}{m+n-j}\binom{k}{j}$. Verified: no Solve time: 4m23s Let $$ A=m-r+s,\qquad B=n+r-s,\qquad M=m+n. $$ The sum is $$ S=\sum_k \binom{A}{k}\binom{B}{n-k}\binom{r+k}{M}. $$ 1. Expand the shifted binomial coefficient Use the given identity $$ \binom{r+k}{M}=\sum_j \binom{r}{M-j}\binom{k}{j}. $$...
TAOCP 1.2.6 Exercise 28
Section 1.2.6: Binomial Coefficients Exercise 28. [ M25 ] Prove that $$ \sum_k \binom{r+tk}{k}\binom{s-tk}{n-k} = \sum_{k \ge 0} \binom{r+s-k}{n-k} t^k, $$ if $n$ is a nonnegative integer. Verified: no Solve time: 4m55s Let $$ L_n(r,s,t)=\sum_{k\ge 0} \binom{r+tk}{k}\binom{s-tk}{n-k},\qquad R_n(r,s,t)=\sum_{k\ge 0} \binom{r+s-k}{n-k}t^k. $$ We prove $L_n=R_n$ for all nonnegative integers $n$, using a double induction on $n$ and $s$. 1. Initial condition $n=0$ For $n=0$, only the term $k=0$ contributes in both...
TAOCP 1.2.6 Exercise 22
Section 1.2.6: Binomial Coefficients Exercise 22. [ M20 ] Prove Eq. (26) for the special case $s=(n-1)r+nt$. Verified: yes Solve time: 1m17s Solution Equation (26) is the binomial convolution identity $$ \binom{r+s}{n}=\sum_{k=0}^{n}\binom{r}{k}\binom{s}{n-k}, $$ which follows by comparing coefficients of $x^n$ in two expansions of $(1+x)^{r+s}$. From the binomial theorem (13), $$ (1+x)^{r+s}=\sum_{n\ge 0}\binom{r+s}{n}x^n. $$ Also, $$ (1+x)^{r+s}=(1+x)^r(1+x)^s. $$ Expanding both factors by (13), $$ (1+x)^r(1+x)^s =\left(\sum_{k\ge 0}\binom{r}{k}x^k\right)\left(\sum_{j\ge 0}\binom{s}{j}x^j\right). $$ Multiplying...
TAOCP 1.2.6 Exercise 19
Section 1.2.6: Binomial Coefficients Exercise 19. [ M18 ] Prove Eq. (23) by induction. Verified: yes Solve time: 1m19s Solution Let $$ S_n = \sum_{k=0}^{n} (-1)^k \binom{r}{k}. $$ Eq. (23) states $$ S_n = (-1)^n \binom{r-1}{n}, \qquad n \ge 0. $$ Induction is taken on $n$. For $n=0$, $$ S_0 = \binom{r}{0} = 1, $$ and $$ (-1)^0 \binom{r-1}{0} = 1. $$ Hence the statement holds for $n=0$. Assume the...
TAOCP 1.2.6 Exercise 20
Section 1.2.6: Binomial Coefficients Exercise 20. [ M20 ] Prove Eq. (24) by using Eqs. (21) and (19), then show that another use of Eq. (19) yields Eq. (25). Verified: yes Solve time: 1m11s Solution Start from Eq. (21), the Chu–Vandermonde formula, $$ \sum_{k}\binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n}. $$ Set $n=r+m$, where $m$ is an integer. Substitution gives $$ \sum_{k}\binom{r}{k}\binom{s}{r+m-k}=\binom{r+s}{r+m}. $$ This is Eq. (24). To obtain Eq. (25), apply Eq. (19) to the...
TAOCP 1.2.6 Exercise 17
Section 1.2.6: Binomial Coefficients Exercise 17. [ M18 ] Prove the Chu--Vandermonde formula (21) from Eq. (15), using the idea that $(1+x)^{r+s} = (1+x)^r(1+x)^s$. Verified: yes Solve time: 1m05s Solution Equation (15) gives the binomial expansion $$ (1+x)^r = \sum_{k} \binom{r}{k} x^k, $$ valid for all real $r$ and integer $k \ge 0$, and similarly $$ (1+x)^s = \sum_{j} \binom{s}{j} x^j. $$ Multiplying these two series yields $$ (1+x)^r(1+x)^s =...
TAOCP 1.2.6 Exercise 14
Section 1.2.6: Binomial Coefficients Exercise 14. [ M21 ] Evaluate $\sum_{k=0}^{n} k^4$. Verified: yes Solve time: 1m03s Solution From Eq. (3) we write powers in terms of falling factorials $k^{\underline{m}}$ using Stirling numbers of the second kind: $$ k^4 = k^{\underline{4}} + 6k^{\underline{3}} + 7k^{\underline{2}} + k^{\underline{1}}. $$ Using $k^{\underline{m}} = m!\binom{k}{m}$, this becomes $$ k^4 = 24\binom{k}{4} + 36\binom{k}{3} + 14\binom{k}{2} + \binom{k}{1}. $$ Summing from $k=0$ to $n$...
TAOCP 1.2.6 Exercise 11
Section 1.2.6: Binomial Coefficients Exercise 11. [ M20 ] (E. Kummer, 1852.) Let $p$ be prime. Show that if $p^n$ divides $\binom{a+b}{a}$ but $p^{n+1}$ does not, then $n$ is equal to the number of carries that occur when $a$ is added to $b$ in the $p$-ary number system. Verified: yes Solve time: 1m05s Solution Let $p$ be prime and write the base-$p$ expansions $$ a = \sum_{i \ge 0} a_i...
TAOCP 1.2.6 Exercise 12
Section 1.2.6: Binomial Coefficients Exercise 12. [ M22 ] Are there any positive integers $n$ for which all the nonzero entries in the $n$th row of Pascal's triangle are odd? If so, find all such $n$. Verified: yes Solve time: 1m02s Solution Let $n$ be a positive integer and consider the $n$th row of Pascal’s triangle, consisting of the numbers $\binom{n}{k}$ for $0 \le k \le n$. Since $n$ is...
TAOCP 1.2.6 Exercise 10
Section 1.2.6: Binomial Coefficients Exercise 10. [ M25 ] If $p$ is prime, show that: $\binom{n}{p} \equiv \lfloor n/p \rfloor \pmod p$. $\binom{p}{k} \equiv 0 \pmod p$, for $1 \le k \le p-1$. $\binom{p-1}{k} \equiv (-1)^k \pmod p$, for $0 \le k \le p-1$. $\binom{p+1}{k} \equiv 0 \pmod p$, for $2 \le k \le p-1$. (E. Lucas, 1877.) $$ \binom{n}{k} \equiv \binom{\lfloor n/p \rfloor}{\lfloor k/p \rfloor} \binom{n \bmod p}{k \bmod...
TAOCP 1.2.4 Exercise 43
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 43. [ M23 ] Evaluate $\sum_{k=1}^n \lfloor \sqrt{k} \rfloor$. Verified: yes Solve time: 1m14s Solution Let $m=\lfloor \sqrt{n}\rfloor$. Then $m^2 \le n < (m+1)^2$, so for each integer $j$ with $1 \le j \le m-1$ the inequality $j^2 \le k \le (j+1)^2-1$ implies $\lfloor \sqrt{k} \rfloor = j$, and this interval contains exactly $(j+1)^2 - j^2 = 2j+1$ integers $k$. For...
TAOCP 1.2.4 Exercise 44
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 44. [ M24 ] Show that $$\sum_{k\ge0}\sum_{1 \le j < b} \left\lfloor \frac{n+j b^k}{b^{k+1}} \right\rfloor = n,$$ if $b$ and $n$ are integers, $n \ge 0$, and $b \ge 2$. What is the value of this sum when $n<0$? Verified: yes Solve time: 2m50s The positive case is correct, but the negative case fails because the telescoping argument silently assumes that...
TAOCP 1.2.4 Exercise 42
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 42. [ M24 ] (a) Prove that $$\sum_{k=1}^n a_k = na_n - \sum_{k=1}^{n-1} k(a_{k+1}-a_k),\qquad \text{if } n>0.$$ (b) The preceding formula is useful for evaluating certain sums involving the floor function. Prove that, if $b$ is an integer $\ge 2$, $$ \sum_{k=1}^n \lfloor \log_b k \rfloor = (n+1)\lfloor \log_b n \rfloor - \frac{b^{\lfloor \log_b n \rfloor +1}-b}{b-1}. $$ Verified: yes Solve...
TAOCP 1.2.4 Exercise 41
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 41. [ M23 ] Let $a_1,a_2,a_3,\ldots$ be the sequence $1,2,2,3,3,3,4,4,4,4,\ldots$; find an expression for $a_n$ in terms of $n$, using the floor and/or ceiling function. Verified: yes Solve time: 1m16s Solution Let $T_k = \frac{k(k+1)}{2}$ for $k \ge 1$. The sequence $1,2,2,3,3,3,4,4,4,4,\ldots$ is characterized by the property that the value $k$ occurs exactly for those indices $n$ satisfying $T_{k-1} < n...
TAOCP 1.2.4 Exercise 36
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 36. [ M23 ] Prove that $\sum_{k=1}^n \lfloor k/2 \rfloor = \lfloor n^2/4 \rfloor$; also evaluate $\sum_{k=1}^n \lceil k/2 \rceil$. Verified: yes Solve time: 1m04s Solution Let $n \in \mathbb{Z}$, $n \ge 1$. 1. Evaluation of $\sum_{k=1}^n \left\lfloor \frac{k}{2} \right\rfloor$ Write $n$ in the form $n=2m$ or $n=2m+1$, where $m \ge 0$. Case $n=2m$ Split the sum into pairs $(2i-1,2i)$ for...
TAOCP 1.2.4 Exercise 35
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 35. ▶ [ M20 ] Given that $m$ and $n$ are integers and $n>0$, prove that $$\left\lfloor \frac{x+m}{n} \right\rfloor = \left\lfloor \frac{\lfloor x \rfloor + m}{n} \right\rfloor$$ for all real $x$. (When $m=0$, we have an important special case.) Does an analogous result hold for the ceiling function? Verified: yes Solve time: 1m21s Solution Write $x = \lfloor x \rfloor +...
TAOCP 1.2.4 Exercise 33
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 33. [ M18 ] Given that $m$ and $n$ are integers, evaluate (a) $\left\lfloor \tfrac12(n+m)\right\rfloor + \left\lfloor \tfrac12(n-m+1)\right\rfloor$; (b) $\left\lceil \tfrac12(n+m)\right\rceil + \left\lceil \tfrac12(n-m+1)\right\rceil$. (The special case $m=0$ is worth noting.) Verified: yes Solve time: 1m28s Solution Let $n,m \in \mathbb{Z}$. (a) Write $n+m = 2a + r_1,\quad r_1 \in {0,1},\quad a = \left\lfloor \frac{n+m}{2} \right\rfloor.$ Write also $n-m+1 = 2b...
TAOCP 1.2.4 Exercise 34
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 34. ▶ [ M21 ] What conditions on the real number $b>1$ are necessary and sufficient to guarantee that $\lfloor \log_b x \rfloor = \lfloor \log_b \lfloor x \rfloor \rfloor$ for all real $x \ge 1$? Verified: yes Solve time: 2m35s We determine all real bases $b>1$ for which $$ \lfloor \log_b x \rfloor = \lfloor \log_b \lfloor x \rfloor \rfloor...
TAOCP 1.2.4 Exercise 31
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 31. [ M22 ] Prove that if $f(n)$ is multiplicative, so is $$g(n)=\sum_{d \mid n} f(d).$$ Verified: yes Solve time: 1m03s Solution Let $r \perp s$. Write $$ g(rs)=\sum_{d \mid rs} f(d). $$ Every positive divisor $d$ of $rs$ can be expressed uniquely in the form $d=ab$, where $a \mid r$ and $b \mid s$. To see existence, let $d \mid...
TAOCP 1.2.4 Exercise 29
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 29. [ M22 ] A function $f(n)$ of positive integers $n$ is called multiplicative if $f(rs)=f(r)f(s)$ whenever $r \perp s$. Show that each of the following functions is multiplicative: (a) $f(n)=n^c$, where $c$ is any constant; (b) $f(n)=[n \text{ is not divisible by } k^2 \text{ for any integer } k>1]$; (c) $f(n)=c^k$, where $k$ is the number of distinct primes...
TAOCP 1.2.4 Exercise 32
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 32. [ M18 ] Prove the double-summation identity $$\sum_{d \mid n}\sum_{c \mid d} f(c,d)=\sum_{c \mid n}\sum_{d \mid (n/c)} f(c,cd),$$ for any function $f(x,y)$. Verified: yes Solve time: 1m21s Solution All sums are finite since the set of positive divisors of $n$ is finite. Consider the left-hand side $$ \sum_{d \mid n}\sum_{c \mid d} f(c,d). $$ The indexing set consists of all...
TAOCP 1.2.4 Exercise 28
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 28. ▶ [ M25 ] Show that the method used to prove Theorem F can be used to prove the following extension, called Euler’s theorem : $a^{\varphi(m)} \equiv 1$ (modulo $m$), for any positive integer $m$, when $a \perp m$. (In particular, the number $n'$ in exercise 19 may be taken to be $n^{\varphi(m)-1} \bmod m$.) Verified: yes Solve time: 1m13s...
TAOCP 1.2.4 Exercise 24
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 24. ▶ [ M20 ] To what extent can Laws A, B, C, and D be generalized to apply to arbitrary real numbers instead of integers? Verified: yes Solve time: 3m03s Let congruence for real numbers be defined by $$ x \equiv y \pmod m \quad \Longleftrightarrow \quad x-y \in m\mathbb{Z}, \quad m \neq 0, $$ where $m\mathbb{Z}={mk : k\in\mathbb{Z}}$. This...
TAOCP 1.2.4 Exercise 21
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 21. [ M22 ] ( Fundamental theorem of arithmetic. ) Use Law B and exercise 1.2.1–5 to prove that every integer $n>1$ has a unique representation as a product of primes (except for the order of the factors). In other words, show that there is exactly one way to write $$n=p_1p_2\cdots p_k,$$ where each $p_j$ is prime and $p_1 \le p_2...
TAOCP 1.2.3 Exercise 45
Section 1.2.3: Sums and Products Exercise 45. ▶ [ M25 ] A Hilbert matrix , sometimes called an $n \times n$ segment of the (infinite) Hilbert matrix, is a matrix for which $a_{ij} = 1/(i + j - 1)$. Show that this is a special case of Cauchy’s matrix, find its inverse, show that each element of the inverse is an integer, and show that the sum of all elements...
TAOCP 1.2.3 Exercise 42
Section 1.2.3: Sums and Products Exercise 42. [ M18 ] What is the sum of all $n^2$ elements in the inverse of the combinatorial matrix? Verified: yes Solve time: 1m11s Solution By Exercise 39, the inverse of the combinatorial matrix has entries $$ b_{ij}=\frac{-y+\delta_{ij}(x+ny)}{x(x+ny)}. $$ The sum of all $n^2$ entries is therefore $$ \sum_{i=1}^n\sum_{j=1}^n b_{ij} \frac1{x(x+ny)} \sum_{i=1}^n\sum_{j=1}^n \left(-y+\delta_{ij}(x+ny)\right). $$ By Eq. (8), $$ \sum_{i=1}^n\sum_{j=1}^n \left(-y+\delta_{ij}(x+ny)\right) \sum_{i=1}^n\sum_{j=1}^n (-y) + (x+ny)...
TAOCP 1.2.3 Exercise 43
Section 1.2.3: Sums and Products Exercise 43. [ M24 ] What is the sum of all $n^2$ elements in the inverse of Vandermonde’s matrix? [Hint: Use exercise 33.] Verified: yes Solve time: 1m22s Solution Let $V$ be the Vandermonde matrix with entries $V_{ij} = x_i^{j-1}$ for $1 \le i, j \le n$, and let $B = V^{-1}$ with entries $b_{ij}$. The required sum is $$ S = \sum_{i=1}^n \sum_{j=1}^n b_{ij}....
TAOCP 1.2.3 Exercise 38
Section 1.2.3: Sums and Products Exercise 38. ▶ [ M25 ] Show that the determinant of Cauchy’s matrix is $$\prod_{1 \le i < j \le n} (x_j - x_i)(y_j - y_i) \bigg/ \prod_{1 \le i, j \le n} (x_i + y_j).$$ Verified: yes Solve time: 3m37s Let $$ D_n=\det!\left(\frac{1}{x_i+y_j}\right) {1\le i,j\le n}, \qquad F_n=D_n\prod {1\le i,j\le n}(x_i+y_j). $$ From the original argument (which is correct), we already know: $F_n$ is...
TAOCP 1.2.3 Exercise 39
Section 1.2.3: Sums and Products Exercise 39. [ M23 ] Show that the inverse of a combinatorial matrix is a combinatorial matrix with the entries $b_{ij} = (-y + \delta_{ij}(x + ny)) / x(x + ny)$. Verified: no Solve time: 2m18s Solution Let $A = (a_{ij})_{1 \le i,j \le n}$ be the combinatorial matrix with entries $a_{ij} = x + y,[i \ne j],$ equivalently $a_{ij} = y + x,\delta_{ij}.$ Let...
TAOCP 1.2.3 Exercise 40
Section 1.2.3: Sums and Products Exercise 40. [ M24 ] Show that the inverse of Vandermonde’s matrix is given by $$b_{ij} = \left( \sum_{\substack{1 \le k_1 < \cdots < k_{n-j} \le n \ k_1, \ldots, k_{n-j} \ne i}} (-1)^{j-1} x_{k_1} \ldots x_{k_{n-j}} \right) \bigg/ x_i \prod_{\substack{1 \le k \le n \ k \ne i}} (x_k - x_i).$$ Don’t be dismayed by the complicated sum in the numerator, it is just...
TAOCP 1.2.3 Exercise 36
Section 1.2.3: Sums and Products Exercise 36. [ M23 ] Show that the determinant of the combinatorial matrix is $x^{n-1}(x + ny)$. Verified: yes Solve time: 1m30s Solution Let $A = (a_{ij})_{1 \le i,j \le n}$ be the combinatorial matrix defined by $a_{ij} = y + \delta_{ij} x,$ so that $a_{ii} = x+y$ and $a_{ij} = y$ when $i \ne j$. Define the vector $e = (1,1,\dots,1)^T \in \mathbb{R}^n$. For...
TAOCP 1.2.3 Exercise 37
Section 1.2.3: Sums and Products Exercise 37. ▶ [ M24 ] Show that the determinant of Vandermonde’s matrix is $$\prod_{1 \le j \le n} x_j \prod_{1 \le i < j \le n} (x_j - x_i).$$ Verified: yes Solve time: 1m24s Solution Let $A = (a_{ij})$ be the $n \times n$ matrix defined by $a_{ij} = x_j^i$. Define a second matrix $B = (b_{ij})$ by $b_{ij} = x_j^{,i-1}$ for $1 \le...
TAOCP 1.2.3 Exercise 34
Section 1.2.3: Sums and Products Exercise 34. [ M25 ] Prove that $$\sum_{k=1}^n \frac{\prod_{1 \le r \le n, , r \ne m} (x + k - r)}{\prod_{1 \le r \le n, , r \ne k} (k - r)} = 1,$$ provided that $1 \le m \le n$ and $x$ is arbitrary. For example, if $n = 4$ and $m = 2$, then $$\frac{x(x-2)(x-3)}{(-1)(-2)(-3)} + \frac{(x+1)(x-1)(x-2)}{(1)(-1)(-2)} + \frac{(x+2)x(x-1)}{(2)(1)(-1)} + \frac{(x+3)(x+1)x}{(3)(2)(1)} =...
TAOCP 1.2.3 Exercise 32
Section 1.2.3: Sums and Products Exercise 32. [ M20 ] Prove that $$\prod_{j=1}^n \sum_{i=1}^m a_{ij} = \sum_{1 \le i_1, \ldots, i_n \le m} a_{i_1 1} \ldots a_{i_n n}.$$ Verified: yes Solve time: 1m06s Solution For each integer $j$ with $1 \le j \le n$, define $$ S_j = \sum_{i=1}^m a_{ij}. $$ The left-hand side is $$ \prod_{j=1}^n S_j = S_1 S_2 \cdots S_n. $$ The proof proceeds by repeated use...
TAOCP 1.2.3 Exercise 31
Section 1.2.3: Sums and Products Exercise 31. [ M20 ] Use Binet’s formula to express the sum $\sum_{1 \le j < k \le n} (u_j - u_k)(v_j - v_k)$ in terms of $\sum_{j=1}^n u_j v_j$, $\sum_{j=1}^n u_j$, and $\sum_{j=1}^n v_j$. Verified: yes Solve time: 1m13s Solution Let $$ T = \sum_{1 \le j < k \le n} (u_j - u_k)(v_j - v_k). $$ The summand is symmetric in the sense...
TAOCP 1.2.3 Exercise 28
Section 1.2.3: Sums and Products Exercise 28. [ M22 ] Find a simple formula for $\prod_{j=2}^n (1 - 1/j^2)$. Verified: yes Solve time: 1m14s Solution For $j \ge 2$, $$ 1 - \frac{1}{j^2} = \frac{j^2 - 1}{j^2} = \frac{(j-1)(j+1)}{j^2} = \frac{j-1}{j} \cdot \frac{j+1}{j}. $$ Hence $$ \prod_{j=2}^n \left(1 - \frac{1}{j^2}\right) = \prod_{j=2}^n \frac{j-1}{j} \cdot \prod_{j=2}^n \frac{j+1}{j}. $$ First product: $$ \prod_{j=2}^n \frac{j-1}{j} = \frac{1}{2} \cdot \frac{2}{3} \cdots \frac{n-1}{n}. $$ All...
TAOCP 1.2.3 Exercise 30
Section 1.2.3: Sums and Products Exercise 30. ▶ [ M23 ] (J. Binet, 1812.) Without using induction, prove the identity $$\left( \sum_{j=1}^n a_j x_j \right) \left( \sum_{j=1}^n b_j y_j \right) = \left( \sum_{j=1}^n a_j y_j \right) \left( \sum_{j=1}^n b_j x_j \right) + \sum_{1 \le j < k \le n} (a_j b_k - a_k b_j)(x_j y_k - x_k y_j).$$ [An important special case arises when $w_1, \ldots, w_n, z_1, \ldots, z_n$...
TAOCP 1.2.3 Exercise 27
Section 1.2.3: Sums and Products Exercise 27. [ M20 ] Generalize the result of exercise 1.2.1–9 by proving that $$\prod_{j=1}^n (1 - a_j) \ge 1 - \sum_{j=1}^n a_j,$$ assuming that $0 < a_j < 1$. Verified: yes Solve time: 1m04s Solution Let $P_n = \prod_{j=1}^n (1 - a_j)$ and $S_n = \sum_{j=1}^n a_j$, where $0 < a_j < 1$ for each $j$. For $n = 1$, the identity reduces to...
TAOCP 1.2.3 Exercise 21
Section 1.2.3: Sums and Products Exercise 21. ▶ [ M25 ] Derive rule (d) from (8) and (17). Verified: yes Solve time: 1m11s Solution From (17), each restricted sum may be written as a full sum using Iverson’s convention: $$ \sum_{R(j)} a_j = \sum_j a_j [R(j)], \qquad \sum_{S(j)} a_j = \sum_j a_j [S(j)]. $$ Hence $$ \sum_{R(j)} a_j + \sum_{S(j)} a_j = \sum_j a_j [R(j)] + \sum_j a_j [S(j)]. $$...
TAOCP 1.2.3 Exercise 18
Section 1.2.3: Sums and Products Exercise 18. [ M20 ] Show how to interchange the order of summation as in Eq. (9) given that $R(i)$ is the relation “$n$ is a multiple of $i$” and $S(i, j)$ is the relation “$1 \le j < i$.” Verified: yes Solve time: 1m13s Solution Let $R(i)$ be the relation “$i$ is a positive integer such that $n$ is a multiple of $i$,” equivalently...
TAOCP 1.2.3 Exercise 16
Section 1.2.3: Sums and Products Exercise 16. [ M22 ] Prove that $$\sum_{j=0}^n jx^j = \frac{nx^{n+2} - (n+1)x^{n+1} + x}{(x-1)^2},$$ if $x \ne 1$, without using mathematical induction. Verified: yes Solve time: 1m20s Solution Let $$ S=\sum_{j=0}^n jx^j. $$ Multiply both sides by $x$: $$ xS=\sum_{j=0}^n jx^{j+1}. $$ Change index by writing the sum in powers of $x^j$: $$ xS=\sum_{j=1}^{n+1} (j-1)x^j. $$ The term corresponding to $j=0$ is absent since...
TAOCP 1.2.3 Exercise 15
Section 1.2.3: Sums and Products Exercise 15. ▶ [ M22 ] Compute the sum $1 \times 2 + 2 \times 2^2 + 3 \times 2^3 + \cdots + n \times 2^n$ for small values of $n$. Do you see the pattern developing in these numbers? If not, discover it by manipulations similar to those leading up to Eq. (14). Verified: yes Solve time: 1m14s Solution Let $$ S_n = \sum_{k=1}^{n}...
TAOCP 1.2.2 Exercise 27
Section 1.2.2: Numbers, Powers, and Logarithms Exercise 27. ▶ [ M25 ] Consider the method for calculating $\log_{10} x$ discussed in the text. Let $x' k$ denote the computed approximation to $x_k$, determined as follows: $x(1 - \delta) \le 10^n x' 0 \le x(1 + \epsilon)$; and in the determination of $x' k$ by Eqs. (18), the quantity $y_k$ is used in place of $(x' {k-1})^2$, where $(x' {k-1})^2(1 -...