An improved bound on Boolean matrix multiplication for highly clustered data

Leszek Ga̧sieniec, Andrzej Lingas

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Scopus citations

Abstract

We consider the problem of computing the product of two n x n Boolean matrices A and B. For two 0-1 strings s = s1s2....sm and u = u1u2...um, an extended Hamming distance, eh(s, u), between the strings, is defined by a recursive equation eh(s,u) = eh(sl+1...sm,ul+1...um) + (s1 + u1 mod 2), where l is the maximum number, s.t., sj = s1 and uj = u1 for j = 1,...,l. For any n x n Boolean matrix C, let GC be a complete weighted graph on the rows of C, where the weight of an edge between two rows is equal to its extended Hamming distance. Next, let MWT(C) be the weight of a minimum weight spanning tree of GC. We show that the product of A and B as well as the so called witnesses of the product can be computed in time Ō(n(n + min{MWT(A), MWT(Bt)})) 1. Since the extended Hamming distance between two strings never exceeds the standard Hamming distance between them, our result subsumes an earlier similar result on the Boolean matrix product in terms of the Hamming distance due to Björklund and Lingas [4]. We also observe that min{MWT(A), MWT(Bt)} = O(min{rA,rB}), where rA and TB reflect the minimum number of rectangles required to cover 1s in A and B, respectively. Hence, our result also generalizes the recent upper bound on the Boolean matrix product in terms of rA and rB, due to Lingas [12].

Original languageEnglish (US)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
EditorsFrank Dehne, Jorg-Rudiger Sack, Michiel Smid
PublisherSpringer Verlag
Pages329-339
Number of pages11
ISBN (Print)3540405453
DOIs
StatePublished - Jan 1 2003
Externally publishedYes

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2748
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Ga̧sieniec, L., & Lingas, A. (2003). An improved bound on Boolean matrix multiplication for highly clustered data. In F. Dehne, J-R. Sack, & M. Smid (Eds.), Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp. 329-339). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2748). Springer Verlag. https://doi.org/10.1007/978-3-540-45078-8_29