Faster generation of random spanning trees

Abstract

In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative (1+ d) of uniform in expected time O[over-(m[sqrt]n log 1/d).This improves the sparse graph case of the best previously known worst-case bound of O(min {mn, n[superscript 2.376]}),which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks, and we use this to introduce a new approach to the problem that integrates discrete random walk-based techniques with continuous linear algebraic methods. We believe that our use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is a useful paradigm that will find further applications in algorithmic graph theory.