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Dijkstra, Floyd and Warshall meet Kleene

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Formal Aspects of Computing

Abstract

Around 1960, Dijkstra, Floyd and Warshall published papers on algorithms for solving single-source and all-sources shortest path problems, respectively. These algorithms, nowadays named after their inventors, are well known and well established. This paper sheds an algebraic light on these algorithms. We combine the shortest path problems with Kleene algebra, also known as Conway’s regular algebra. This view yields a purely algebraic version of Dijkstra’s shortest path algorithm and the one by Floyd/Warshall. Moreover, the algebraic abstraction yields applications of these algorithms to structures different from graphs and pinpoints the mathematical requirements on the underlying cost algebra that ensure their correctness.

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Authors and Affiliations

  1. NICTA, Sydney, Australia

    Peter Höfner

  2. University of New South Wales, Sydney, Australia

    Peter Höfner

  3. Universität Augsburg, Augsburg, Germany

    Bernhard Möller

Authors
  1. Peter Höfner
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  2. Bernhard Möller
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Corresponding author

Correspondence to Bernhard Möller.

Additional information

by Eerke Boiten and Jim Woodcock

It is our pleasure to dedicate this paper to Charles Carroll Morgan on the occasion of his 60th birthday.With this paperwe try to combine two of Carroll’s various research interests: Programming Methodology and Algorithmic Calculi. The former includes problem-solving techniques, such as algorithms, the latter includes the study of calculation of programs from specifications, in particular, formal refinement techniques. We illustrate this with a fresh look at path problems in graphs, a topic which is regaining interest in connection with protocol verification, another of Carroll’s interests.

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Höfner, P., Möller, B. Dijkstra, Floyd and Warshall meet Kleene. Form Asp Comp 24, 459–476 (2012). https://doi.org/10.1007/s00165-012-0245-4

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  • DOI: https://doi.org/10.1007/s00165-012-0245-4

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