LATIN 2016: LATIN 2016: Theoretical Informatics pp 522-535

# Generating Random Spanning Trees via Fast Matrix Multiplication

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

## 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.

## Keywords

Uniform spanning trees Spectral graph theory Fast matrix multiplication Laplacian matrices

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