Beyond the worst-case bisection bound: Fast sorting and ranking on meshes

Michael Kaufmann, Jop F. Sibeyn, Torsten Suel

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Sorting is an important subroutine in many parallel algorithms and has been studied extensively on meshes and related networks. If every processor of an n × n mesh is the source and destination of at most k elements, then sorting requires at least k k; n/2 steps in the worst-case, and simple algorithms have recently been proposed that nearly match this bound. However, this lower bound does not extend to non-worst-case inputs, or weaker definitions of sorting that are sufficient in many applications. In this paper, we give algorithms and lower bounds for several such problems. We first present a very simple scheme for k-k routing that performs optimally under both average-case and worst-case inputs. As an application of this scheme, we describe a simple k-k sorting algorithm based on sample sort that nearly matches this bound. The main part of the paper considers several 'sorting-like' problems. In the ranking problem, the ranks of all elements have to be determined, but there is no requirement about their final positions. We describe an algorithm running in time (1 +o(l)) k n/4 steps, which is nearly optimal under the considered model of the mesh. We show that integer versions of the sorting and ranking problems, where keys are drawn from {0,…, m — 1}, can be solved asymptotically faster than the general problems for small values of m. A related problem, the excess counting problem, can be solved in O(n) steps in many interesting cases.

    Original languageEnglish (US)
    Title of host publicationAlgorithms - ESA 1995 - 3rd Annual European Symposium, Proceedings
    EditorsPaul Spirakis
    PublisherSpringer Verlag
    Pages75-88
    Number of pages14
    ISBN (Print)3540603131, 9783540603139
    DOIs
    StatePublished - 1995
    Event3rd Annual European Symposium on Algorithms, ESA 1995 - Corfu, Greece
    Duration: Sep 25 1995Sep 27 1995

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume979
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other3rd Annual European Symposium on Algorithms, ESA 1995
    CountryGreece
    CityCorfu
    Period9/25/959/27/95

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computer Science(all)

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  • Cite this

    Kaufmann, M., Sibeyn, J. F., & Suel, T. (1995). Beyond the worst-case bisection bound: Fast sorting and ranking on meshes. In P. Spirakis (Ed.), Algorithms - ESA 1995 - 3rd Annual European Symposium, Proceedings (pp. 75-88). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 979). Springer Verlag. https://doi.org/10.1007/3-540-60313-1_135