Generating Random Spanning Trees via Fast Matrix Multiplication

  • Nicholas J. A. Harvey
  • Keyulu XuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


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.


Uniform spanning trees Spectral graph theory Fast matrix multiplication Laplacian matrices 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada

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