Abstract

This paper presents new algorithms for the parallel evaluation of certain polynomial expression. In particular, for the parallel evaluation of xn,we introduce an algorithm which takes two steps of parallel division and [log2n] steps of parallel addition, while the usual algorithm takes [log2n] steps of parallel multiplication. Hence our algorithm is faster than the usual algorithms when multiplication takes more time than addition. Similar algorithm for the evaluation of other polynomial expressions are also introduced. Lower bounds on the time needed for the parallel evaluation of rational expressions are given. All the algorithms presented in the paper are shown to be asymptotically optimal. Moreover, we prove that by using parallelism the evaluation of any first order rational recurrence, e.g., [equation], and any non-linear polynomial recurrence can be sped up at most by a constant factor, no matter how many processors are used.

Highlights

  • In this paper we consider the parallel evaluation of certain rational expressions

  • By using parallelism, the evaluation of an expression defined by any first-order rational recurrence of degree greater than 1 or any nonlinear polynomial recurrence can be sped up at most by a constant factor, no matter how many processors are used

  • By the construction of the tree, the rational expressions associated with internal nodes are not in R~. (It is clear that the tree is finite, since there is a positive lower bound on the time needed for every operation.) We note that if the binary operation associated with an internal node is a nonscalar addition, multiplication, or division the two successors of the node must be leaves

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Summary

Introduction

In this paper we consider the parallel evaluation of certain rational expressions. This assumption corresponds to multiple-instruction streammultiple-data stream ( M I M D ) machines (Flynn [4]) such as C.mmp, the multi-miniprocessor system at Carnegie-Mellon University (Wulf and Bell [19]).

Lower Bounds
Results on Nonlinear Recurrence Problems
Summary and Conclusions
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