Generating Random Spanning Trees via Fast Matrix Multiplication

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 nvertex graph in time O(n2.38 ). 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(n2.38 ) 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.

• Type: Journal
• Year: 2016
• Where: LATIN
• Authors: Nicholas J. A. Harvey, Keyulu Xu
• Keywords: Information Technology, LATIN 2016