Encyclopedia of Algorithms

2008 Edition
| Editors: Ming-Yang Kao

All Pairs Shortest Paths via Matrix Multiplication

2002; Zwick
  • Tadao Takaoka
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30162-4_12

Keywords and Synonyms

Shortest path problem; Algorithm analysis          

Problem Definition

The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with non-negative real numbers as edge costs. Focus is given on shortest distances between vertices, as shortest paths can be obtained with a slight increase of cost. Classically, the APSP problem can be solved in cubic time of O(n3). The problem here is to achieve a sub-cubic time for a graph with small integer costs.

A directed graph is given by \( { G=(V,E) } \)

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Recommended Reading

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    Alon, N., Galil, Z., Margalit, O.: On the exponent of the all pairs shortest path problem. In: Proc. 32th IEEE FOCS, pp. 569–575. IEEE Computer Society, Los Alamitos, USA (1991). Also JCSS 54, 255–262 (1997)Google Scholar
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    Alon, N., Galil, Z., Margalit, O., Naor, M.: Witnesses for Boolean matrix multiplication and for shortest paths. In: Proc. 33th IEEE FOCS, pp. 417–426. IEEE Computer Society, Los Alamitos, USA (1992)Google Scholar
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    Takaoka, T.: Sub-cubic time algorithms for the all pairs shortest path problem. Algorithmica 20, 309–318 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
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    Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49(3), 289–317 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Tadao Takaoka
    • 1
  1. 1.Department of Computer Science and Software EngineeringUniversity of CanterburyChristchurchNew Zealand