Efficient weighted multiselection in parallel architectures

©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

Efficient parallel computation on multiprocessors with optical interconnection networks

This dissertation studies optical interconnection networks, their architecture, address schemes, and computation and communication capabilities. We focus on a simple but powerful optical interconnection network model - the Linear Array with Reconfigurable pipelined Bus System (LARPBS). We extend the LARPBS model to a simplified higher dimensional LAPRBS and provide a set of basic computation operations. We then study the following two groups of parallel computation problems on both one dimensional LARPBS\u27s as well as multi-dimensional LARPBS\u27s: parallel comparison problems, including sorting, merging, and selection; Boolean matrix multiplication, transitive closure and their applications to connected component problems. We implement an optimal sorting algorithm on an n-processor LARPBS. With this optimal sorting algorithm at disposal, we study the sorting problem for higher dimensional LARPBS\u27s and obtain the following results: • An optimal basic Columnsort algorithm on a 2D LARPBS. • Two optimal two-way merge sort algorithms on a 2D LARPBS. • An optimal multi-way merge sorting algorithm on a 2D LARPBS. • An optimal generalized column sort algorithm on a 2D LARPBS. • An optimal generalized column sort algorithm on a 3D LARPBS. • An optimal 5-phase sorting algorithm on a 3D LARPBS. Results for selection problems are as follows: • A constant time maximum-finding algorithm on an LARPBS. • An optimal maximum-finding algorithm on an LARPBS. • An O((log log n)2) time parallel selection algorithm on an LARPBS. • An O(k(log log n)2) time parallel multi-selection algorithm on an LARPBS. While studying the computation and communication properties of the LARPBS model, we find Boolean matrix multiplication and its applications to the graph are another set of problem that can be solved efficiently on the LARPBS. Following is a list of results we have obtained in this area. • A constant time Boolean matrix multiplication algorithm. • An O(log n)-time transitive closure algorithm. • An O(log n)-time connected components algorithm. • An O(log n)-time strongly connected components algorithm. The results provided in this dissertation show the strong computation and communication power of optical interconnection networks

Optimal parallel weighted multiselection

Copyright © 2002 IEEEWeighted multiselection requires us to select r elements from a given set of n elements, each associated with a weight such that each element selected is on a pre-specified weighted-rank, 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 the set is not smaller than k. This paper presents efficient algorithms for solving this problem both sequentially and in parallel on EREW PRAM. Our sequential algorithm solves this problem in time O(nlogr) which is optimal. Our parallel algorithm runs in O(T/sub 1/logr) time on an EREW PRAM with 1 < p /spl les/ n processors, and is optimal with respect to T/sub 1/ which is the time complexity for single-element weighted selection using p processors. We give a parallel algorithm for single-element weighted selection using p EREW processors which runs cost-optimally in O(n/p) time for 1 < p /spl les/ nloglogn/logn, and time-optimally in O(logn/loglogn) time for nloglogn/logn < p /spl les/ n.Hong She