Sampling random spanning trees faster than matrix multiplication

Abstract

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in (n5/3 m1/3) time. The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, O(nω). For the special case of unweighted graphs, this improves upon the best previously known running time of Õ(min{nω,m√n,m4/3}) for m ⪢ n7/4 (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15).