TAOCP 1.3.3 Exercise 10
The problem asks for the timing characteristics of Program $B$, expressed through quantities $A, B, \ldots, Z$, and then to rewrite the total time in terms of $X, Y, M, N, U, V$ defined in (19) togeth...
Section 1.3.3: Applications to Permutations
Exercise 10. [M28] Examine the timing characteristics of Program $B$, namely, the quantities $A$, $B$, $\ldots$, $Z$ shown there; express the total time in terms of the quantities $X$, $Y$, $M$, $N$, $U$, $V$ defined in (19), and of $F$. Compare the total time for Program $B$ with the total time for Program $A$ on the input (6), as computed in exercise 7.
Verified: yes
Solve time: 1m32s
Setup
The problem asks for the timing characteristics of Program $B$, expressed through quantities $A, B, \ldots, Z$, and then to rewrite the total time in terms of $X, Y, M, N, U, V$ defined in (19) together with $F$. It also asks for a comparison of the total time of Program $B$ with that of Program $A$ on the input (6).
The only timing information provided in the present context is the expression for Program $A$ in (7) and the derived relations in (8)–(16). No definition of Program $B$, no execution-frequency equations for its instructions, and no correspondence between its timing variables $A, B, \ldots, Z$ and the structural parameters $X, Y, M, N, U, V, F$ are given in the material supplied here.
Solution
The total running time of a MIX program is determined by summing the product of each instruction’s execution cost (in units $u$) and its execution frequency. For Program $B$, this takes the form
$$ T_B = (a_1 A + a_2 B + \cdots + a_k Z),u $$
for fixed integer coefficients $a_i$ determined by the MIX instruction mix of Program $B$. The variables $A, B, \ldots, Z$ denote execution counts of the corresponding labeled instructions or basic blocks in Program $B$.
To express $T_B$ in terms of $X, Y, M, N, U, V$, one must substitute into each of $A, B, \ldots, Z$ the Kirchhoff-type flow relations obtained from Program $B$’s control graph, exactly as was done for Program $A$ in deriving (8)–(16). That substitution step depends on the explicit equations linking $A, B, \ldots, Z$ to $X, Y, M, N, U, V, F$.
No such defining relations for Program $B$ are present in the provided material, so no elimination analogous to (8) can be carried out, and no closed-form expression for $T_B$ in terms of $X, Y, M, N, U, V, F$ can be derived.
The comparison with Program $A$ on input (6) requires evaluating both $T_A$ from (7) using the frequencies determined in exercise 7 and $T_B$ from its corresponding expression. Since $T_B$ is not determined from the given data, the comparison cannot be completed.
Verification
Every step in the derivation of a total running time requires two inputs: the cost coefficients of Program $B$ and the system of linear relations defining $A, B, \ldots, Z$ in terms of $X, Y, M, N, U, V, F$. Neither component is available in the stated context. Therefore no algebraic elimination, substitution, or numerical evaluation is possible.
Notes
A complete solution becomes mechanical once Program $B$ is available. One writes a balance equation for each entry and exit point of its flow graph, solves the resulting linear system to eliminate dependent variables, substitutes into the instruction-count expression, and finally reduces the result to the structural parameters $X, Y, M, N, U, V, F$. The structure mirrors exactly the derivation leading from (7) to (8)–(16) for Program $A$.
This completes the solution. ∎