Work-time optimal k-merge algorithms on the PRAM

Abstract

For 2/spl les/k/spl les/n, the k-merge problem is to merge a collection of ksorted sequences of total length n into a new sorted sequence. The k-merge problem is fundamental as it provides a common generalization of both merging and sorting. The main contribution of this work is to give simple and intuitive work-time optimal algorithms for the k-merge problem on three PRAM models, thus settling the status of the k-merge problem. We first prove that /spl Omega/(n log k) work is required to solve the k-merge problem on the PRAM models. We then show that the EREW-PRAM and both the CREW-PRAM and the CRCW require /spl Omega/(log n) time and /spl Omega/(log log n+log k) time, respectively, provided that the amount of work is bounded by O(n log k). Our first k-merge algorithm runs in /spl Theta/(log n) time and performs /spl Theta/(n log k) work on the EREW-PRAM. Finally, we design a work-time optimal CREW-PRAM k-merge algorithm that runs in /spl Theta/(log log n+log k) time and performs /spl Theta/(n log k) work. This latter algorithm is also work-time optimal on the CREW-PRAM model. Our algorithms completely settle the status of the k-merge problem on the three main PRAM models.