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A Geometric Approach to Boolean Matrix Multiplication

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Algorithms and Computation (ISAAC 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2518))

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Abstract

For a Boolean matrix D, let r D be the minimum number of rectangles sufficient to cover exactly the rectilinear region formed by the 1-entries in D. Next, let m D be the minimum of the number of 0-entries and the number of 1-entries inD.

Suppose that the rectilinear regions formed by the 1-entries in two n*n Boolean matrices A and B totally with q edges are given. We show that in time Õ(q + minr ArB, n(n + rA),n(n) + rB))1 one can construct a data structure which for any entry of the Boolean product of A and B reports whether or not it is equal to 1, and if so, reports also the so called witness of the entry, in time O(log q). Asa corollary, we infer that if the matrices A and B are given as input, their product and the witnesses of the product can be computed in time O(n (n+minr a,rb)) . This implies in particular that the product of A and B and its witnesses can be computed in time O(n(n + minm A, mb)).

In contrast to the known sub-cubic algorithms for Boolean matrix multiplication based on arithmetic 0 — 1-matrix multiplication, our algorithms do not involve large hidden constants in their running time and are easy to implement.

Research supported in part by TFR grant 221-99-344.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Lund University, Box 118, S-22100, Lund, Sweden

    Andrzej Lingas

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  1. Andrzej Lingas
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Editor information

Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada, K1S 5B6

    Prosenjit Bose  & Pat Morin  & 

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© 2002 Springer-Verlag Berlin Heidelberg

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Lingas, A. (2002). A Geometric Approach to Boolean Matrix Multiplication. In: Bose, P., Morin, P. (eds) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36136-7_44

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  • DOI: https://doi.org/10.1007/3-540-36136-7_44

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00142-3

  • Online ISBN: 978-3-540-36136-7

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