TAOCP 4.3.1 Exercise 30

Section 4.3.1: The Classical Algorithms

Exercise 30. ▶ [**] [22] If memory space is limited, it may be desirable to use the same storage locations for both input and output during the performance of some of the algorithms in this section. Is it possible to have the quotient $q_0, \ldots, q_n$ stored in the same respective locations as $u_0, \ldots, u_{n-1}$ or $v_0, \ldots, v_{n-1}$ during Algorithm A or S? Is it possible to have the quotient $q_0, \ldots, q_m$ occupy the same locations as $u_0, \ldots, u_{m+n}$ in Algorithm D? Is there any permissible overlap of memory locations between input and output in Algorithm M?

Verified: no
Solve time: 2m23s

In Algorithms A and S, the computation of each output digit $w_j$ depends on the corresponding input digits $u_j$ and $v_j$ as well as on the carry or borrow from the previous step. Therefore, storing the quotient or sum/difference in the same locations as the input would overwrite values that are still needed for subsequent computations. Consequently, it is not possible to have the quotient $q_0, \ldots, q_{n-1}$ occupy the same locations as $u_0, \ldots, u_{n-1}$ or $v_0, \ldots, v_{n-1}$ during the execution of Algorithm A or S.

In Algorithm D, each quotient digit $q_j$ is determined from multiple digits of the current remainder $u_{j}, \ldots, u_{j+n}$; overwriting these digits with $q_j$ would destroy information required to compute later quotient digits. Thus the quotient $q_0, \ldots, q_m$ cannot occupy the same locations as $u_0, \ldots, u_{m+n}$. In Algorithm M, the partial products are accumulated in the output array $w$, but the input arrays $u$ and $v$ are accessed in a strictly read-only manner. Consequently, it is permissible for $w$ to overlap either $u$ or $v$, provided that the partial products are added in the correct order from low to high indices, so that no input value is overwritten before it is used. This allows some flexibility in memory usage for multiplication but not for addition, subtraction, or division. ∎