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Generating Random Spanning Trees via Fast Matrix Multiplication

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LATIN 2016: Theoretical Informatics (LATIN 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9644))

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Abstract

We consider the problem of sampling a uniformly random spanning tree of a graph. This is a classic algorithmic problem for which several exact and approximate algorithms are known. Random spanning trees have several connections to Laplacian matrices; this leads to algorithms based on fast matrix multiplication. The best algorithm for dense graphs can produce a uniformly random spanning tree of an n-vertex graph in time \(O(n^{2.38})\). This algorithm is intricate and requires explicitly computing the LU-decomposition of the Laplacian.

We present a new algorithm that also runs in time \(O(n^{2.38})\) but has several conceptual advantages. First, whereas previous algorithms need to introduce directed graphs, our algorithm works only with undirected graphs. Second, our algorithm uses fast matrix inversion as a black-box, thereby avoiding the intricate details of the LU-decomposition.

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Notes

  1. 1.

    \(P := I - \mathbf {1}\mathbf {1}^T/n\)

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Authors and Affiliations

  1. University of British Columbia, Vancouver, BC, Canada

    Nicholas J. A. Harvey & Keyulu Xu

Authors
  1. Nicholas J. A. Harvey
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  2. Keyulu Xu
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Corresponding author

Correspondence to Keyulu Xu .

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Editors and Affiliations

  1. Carleton University, Ottawa, ON, Ontario, Canada

    Evangelos Kranakis

  2. University Chile, Santiago, Chile

    Gonzalo Navarro

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Harvey, N.J.A., Xu, K. (2016). Generating Random Spanning Trees via Fast Matrix Multiplication. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_39

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  • DOI: https://doi.org/10.1007/978-3-662-49529-2_39

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-49528-5

  • Online ISBN: 978-3-662-49529-2

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