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International Colloquium on Theoretical Aspects of Computing

ICTAC 2016: Theoretical Aspects of Computing – ICTAC 2016 pp 51–68Cite as

Relation-Algebraic Verification of Prim’s Minimum Spanning Tree Algorithm

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9965))

Abstract

We formally prove the correctness of Prim’s algorithm for computing minimum spanning trees. We introduce new generalisations of relation algebras and Kleene algebras, in which most of the proof can be carried out. Only a small part needs additional operations, for which we introduce a new algebraic structure. We instantiate these algebras by matrices over extended reals, which model the weighted graphs used in the algorithm. Many existing results from relation algebras and Kleene algebras generalise from the relation model to the weighted-graph model with no or small changes. The overall structure of the proof uses Hoare logic. All results are formally verified in Isabelle/HOL heavily using its integrated automated theorem provers.

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Acknowledgements

I thank the anonymous referees for helpful feedback including the suggestion to generalise Lemma 1 to its present form. I thank Rudolf Berghammer for discussions about the cardinality of relations and ways to generalise it. I thank Peter Höfner and Bernhard Möller for discussing alternative approaches to minimum spanning trees in Kleene algebras. I thank the participants of the 73rd meeting of IFIP WG 2.1, the 14th Logic and Computation Seminar of Kyushu University and the 2016 Workshop on Universal Structures in Mathematics and Computing for the opportunity to talk about this work and for their valuable feedback. The presentation at Kyushu University was part of a JSPS Invitation Fellowship for Research in Japan.

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  1. Department of Computer Science and Software Engineering, University of Canterbury, Christchurch, New Zealand

    Walter Guttmann

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  1. Walter Guttmann
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Correspondence to Walter Guttmann .

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  1. Universidade Federal de Pernambuco Centro de Informática, Recife, Pernambuco, Brazil

    Augusto Sampaio

  2. Deptartment of Electrical Engineering BL, National Taiwan University, Taipei, Taiwan

    Farn Wang

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Guttmann, W. (2016). Relation-Algebraic Verification of Prim’s Minimum Spanning Tree Algorithm. In: Sampaio, A., Wang, F. (eds) Theoretical Aspects of Computing – ICTAC 2016. ICTAC 2016. Lecture Notes in Computer Science(), vol 9965. Springer, Cham. https://doi.org/10.1007/978-3-319-46750-4_4

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  • DOI: https://doi.org/10.1007/978-3-319-46750-4_4

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