Fast Generation of Random Spanning Trees and the Effective Resistance Metric

TL;DR

This paper introduces a faster algorithm for generating uniformly random spanning trees in undirected graphs, leveraging effective resistance and random walks to improve efficiency especially in sparse graphs.

Contribution

The authors develop a novel algorithm that samples random spanning trees in expected oretro{O}(m^{4/3}) time, improving previous bounds by exploiting effective resistance and graph structure.

Findings

Expected runtime oretro{O}(m^{4/3}) for sparse graphs

New connection between effective resistance and graph cuts

Enhanced understanding of random spanning trees

Abstract

We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of $\min(\tilde{O}(m\sqrt{n}),O(n^{\omega}))$ -- that follows from the work of Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse. At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way to algorithmically relate these concepts to the combinatorial structure of the graph. This involves, in particular, establishing a new connection between the effective resistance metric and the cut structure of the underlying graph.