Efficient weighted multiselection in parallel architectures

Abstract

©2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.We study parallel solutions to the problem of weighted multiselection to select r elements on given weighted-ranks from a set S of n weighted elements, where an element is on weighted rank k if it is the smallest element such that the aggregated weight of all elements not greater than it in S is not smaller than k. We propose efficient algorithms on two of the most popular parallel architectures, hypercube and mesh. For a hypercube with p < n processors, we present a parallel algorithm running in 0(n^\varepsilon \min \{ r,\log p\} ) time for p = n^{1 - \varepsilon } ,0 < \varepsilon < 1 which is cost optimal when r \geqslant p. Our algorithm on \sqrt p \times \sqrt p mesh runs in 0(\sqrt p + \frac{n}{p}\log ^3 p) time which is the same as multiselection on mesh when r \geqslant \log p, and thus has the same optimality as multiselection in this case