New Algorithms and Lower Bounds for the Parallel Evaluation of Certain Rational Expressions and Recurrences

H T Kung

doi:10.1145/321941.321945

Abstract

The parallel evaluation of rational expressions is considered. New algorithms which minimize the number of multiplication or division steps are given. They are faster than the usual algorithms when multiplication or division takes more time than addition or subtraction. It is shown, for example, that x n can be evaluated in two steps of parallel division and ⌈log 2 n ⌉ steps of parallel addition, while the usual algorithm takes ⌈log 2 n ⌉ steps of parallel multiplication. Lower bounds on the time required are obtained in terms of the degree of the expressions to be evaluated. From these bounds, the algorithms presented in the paper are shown to be asymptotically optimal. Moreover, it is shown that by using parallelism the evaluation of any first-order rational recurrence of degree greater than 1, e.g. y i +1 = 1/2;( y i + a / y i ), and any nonlinear polynomial recurrence can be sped up at most by a constant factor, no matter how many processors are used and how large the size of the problem is.

Highlights

In this paper we consider the parallel evaluation of certain rational expressions
If we deal with expressions involving, for example, matrices or multiple-precision numbers multiplication is more expensive than addition. (Here we interpret arithmetic operations as matrix or multiple-precision number operations.)
We prove lower bounds on the time needed for the parallel evaluation of certain rational expressions, under the assumption that all processors can perform different operations at any time

Summary

Introduction

]. If division is not used I [log n]M is a lower bound on the time for the parallel evaluation of x n , no matter how many processors are used. In the proof of the following theorem we give an algorithm for the parallel evaluation of xn which uses divisions and which takes time less than [log n] when n is large. If k > n~ the set Ix2 ,x3 ,...,xn~ can be evaluated in five steps of parallel non-scalar multiplication or division and [log n] + 4 steps of parallel addition.

Results

Conclusion