Easy Multiple-Precision Divisors and Word-RAM Constants

Abstract

For integers $b\ge 2$ and $w\ge 1$, define the (b, w) cover size of an integer A as the smallest nonnegative integer k such that A can be written in the form $A=\sum _{i=1}^k(-1)^{\sigma _i}b^{\ell _i}d_i$, where $\sigma _i$, $\ell _i$ and $d_i$ are nonnegative integers and $0\le d_i<b^w$, for $i=1,\ldots ,k$. We study the efficient execution of arithmetic operations on (multiple-precision) integers of small (b, w) cover size on a word RAM with words of w b-ary digits and constant-time multiplication and division. In particular, it is shown that if A is an n-digit integer and B is a nonzero m-digit integer of (b, w) cover size k, then $\lfloor A/B\rfloor$ can be computed in $O(1+{{(k n+m)}/w})$ time. Our results facilitate a unified description of word-RAM algorithms operating on integers that may occupy a fraction of a word or several words.

As an application, we consider the fast generation of integers of a special form for use in word-RAM computation. Many published word-RAM algorithms divide a w-bit word conceptually into equal-sized fields and employ full-word constants whose field values depend in simple ways on the field positions. The constants are either simply postulated or computed with ad-hoc methods. We describe a procedure for obtaining constants of the following form in constant time: The ith field, counted either from the right or from the left, contains g(i), where g is a constant-degree polynomial with integer coefficients that, disregarding mild restrictions, can be arbitrary. This general form covers almost all cases known to the author of word-RAM constants used in published algorithms.