TAOCP 2.2.4 Exercise 10

We wish to adapt Algorithms `A` (addition) and `M` (multiplication) for polynomials in a single variable $x$, allowing exponents up to $b^3 - 1$.

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Section 2.2.4: Circular Lists

Exercise 10. [20] The algorithms in this section assume that we are using three variables $x$, $y$, and $z$ in the polynomials, and that their exponents individually never exceed $b - 1$ (where $b$ is the byte size in MIX's case). Suppose instead that we want to do addition and multiplication of polynomials in only one variable, $x$, and to let its exponent take on values up to $b^3 - 1$. What changes should be made to Algorithms A and M?

Verified: no
Solve time: 39s

Solution

We wish to adapt Algorithms A (addition) and M (multiplication) for polynomials in a single variable $x$, allowing exponents up to $b^3 - 1$. In the original algorithms, each node contains the two-word structure

$$ \text{COEF} \mid ABC, $$

where $ABC$ encodes the triple $(A, B, C)$ of exponents for $(x, y, z)$ in a single unit. For a single variable $x$, we no longer need three separate components; instead, the exponent can be stored directly as a single integer $E$ in one word, taking values $0 \le E \le b^3 - 1$. The list retains the special node with $E = -1$ and $\text{COEF} = 0$ as a sentinel.

Modifications to Algorithm A

Algorithm A relies on comparing ABC(P) and ABC(Q) to locate matching terms. For one-variable polynomials, we replace ABC(P) and ABC(Q) with E(P) and E(Q). The comparison logic remains identical, but is now applied to the single exponent:

  1. Step A2 becomes: if $E(P) < E(Q)$, advance $Q$ along the list; if $E(P) = E(Q)$, go to step A3; if $E(P) > E(Q)$, go to step A5.
  2. Step A3 updates $\text{COEF}(Q) \leftarrow \text{COEF}(Q) + \text{COEF}(P)$. If $\text{COEF}(Q) = 0$, delete the term as in step A4.
  3. Step A5 inserts a new term with exponent $E(P)$ and coefficient $\text{COEF}(P)$.

All pointer manipulations (P, Q, Q1, Q2) remain unchanged. The ordering invariant is now that the list of nodes is in decreasing order of E, with the sentinel node at $E = -1$ linking back to the largest exponent. The rest of the algorithm proceeds without further change.

Modifications to Algorithm M

Algorithm M multiplies polynomials term-by-term, generating new terms whose ABC fields are the componentwise sums of the exponents. For one-variable polynomials, the exponent sum is simply the sum of two single integers:

$$ E(R) \leftarrow E(P) + E(Q), $$

where $P$ iterates over the first polynomial and $Q$ over the second. Since $0 \le E(P), E(Q) \le b^3 - 1$, the maximum exponent is $2 b^3 - 2$. If the word can store integers up to $b^3 - 1$, we must ensure either that arithmetic can accommodate sums up to $2 b^3 - 2$ or that a larger word size is used. Apart from replacing ABC(P) + ABC(Q) with E(P) + E(Q), the insertion into the product polynomial uses the same procedure as in A5, maintaining descending order of exponents. The sentinel node remains with $E = -1$, $\text{COEF} = 0$.

Summary of Changes

  1. Replace the ABC field with a single exponent E of type integer, $0 \le E \le b^3 - 1$.
  2. Replace all comparisons of ABC(P) and ABC(Q) with comparisons of E(P) and E(Q).
  3. In Algorithm M, replace vector addition of ABC triples with ordinary integer addition: E(P) + E(Q).
  4. Maintain descending order of exponents and retain the sentinel node at $E = -1$.
  5. Ensure the word size or arithmetic handles exponents up to $2 b^3 - 2$ if necessary.

No changes are required to pointer handling or the structure of the algorithms beyond these substitutions.

This completes the solution.

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