An almost-linear time algorithm for uniform random spanning tree generation

TL;DR

This paper introduces an almost-linear time algorithm for generating uniform random spanning trees in graphs, significantly improving efficiency over previous methods and applicable to weighted and unweighted graphs.

Contribution

The paper presents the first almost-linear time algorithm for uniform spanning tree generation, utilizing novel Laplacian solver techniques and electrical flow analysis.

Findings

Achieved almost-linear time complexity for uniform spanning tree generation.

Developed a new method using Laplacian solvers to shortcut random walks.

Provided new insights into electrical flows and effective resistance in graphs.

Abstract

We give an $m^{1+o(1)}\beta^{o(1)}$-time algorithm for generating a uniformly random spanning tree in an undirected, weighted graph with max-to-min weight ratio $\beta$. We also give an $m^{1+o(1)}\epsilon^{-o(1)}$-time algorithm for generating a random spanning tree with total variation distance $\epsilon$ from the true uniform distribution. Our second algorithm's runtime does not depend on the edge weights. Our $m^{1+o(1)}\beta^{o(1)}$-time algorithm is the first almost-linear time algorithm for the problem --- even on unweighted graphs --- and is the first subquadratic time algorithm for sparse weighted graphs. Our algorithms improve on the random walk-based approach given in Kelner-M\k{a}dry and M\k{a}dry-Straszak-Tarnawski. We introduce a new way of using Laplacian solvers to shortcut a random walk. In order to fully exploit this shortcutting technique, we prove a number of new facts about electrical flows in graphs. These facts seek to better understand sets of vertices that are well-separated in the effective resistance metric in connection with Schur complements, concentration phenomena for electrical flows after conditioning on partial samples of a random spanning tree, and more.