Richard Cole
Abstract
Megiddo introduced a technique for using a parallel algorithm for one problem to construct an efficient serial algorithm for a second problem. This paper provides a general method that trims a factor of O(log n) time (or more) for many applications of this technique.
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Index Terms
- Slowing down sorting networks to obtain faster sorting algorithms
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Reviews
Daniel P. Bovet
As clearly stated by the author, the paper presents an “improvement of Megiddo's ingenious technique [1], which uses an efficient parallel algorithm for one problem to produce an efficient serial algorithm for a second problem.” In other words, if we want to solve problem B, we look for a closely related problem A such that a serial algorithm solving A in T s steps can easily be adapted to solve B in O( CT s) steps. If we find such A, we use a parallel algorithm to solve A on P processors, serialize it, and show that if B satisfies some requirements, the running time goes down from O( CT pP) to O( CT plog P), if P is not too large. The original technique uses a parallel sorting algorithm to guide a search on a partially ordered space without actually constructing the space. The improvement described in the paper trims a factor of log n from the running time according to the former method. This is obtained by replacing the parallel sorting algorithm with an AKS sorting network [2]. The price to pay for this improvement is twofold: (1) AKS sorting networks involve enormous constants; and (2) the class of problem to which ip applies is more limited since problem A must be a sorting problem. Several geometrically flavored problems are presented that can be solved using the improved technique. The paper is very well written, and it is a pleasure to read. I only have two minor observations: the proof of lemma 1 relies on a formal definipion of sorting networks that is missing; and the equality in lamma 2 is valid when n is a power of 2.
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