Sampling Random Spanning Trees Faster than Matrix Multiplication

TL;DR

This paper introduces a faster algorithm for sampling random spanning trees in weighted graphs, improving previous methods by avoiding matrix multiplication and using Gaussian elimination with effective resistance approximations.

Contribution

The authors develop a novel algorithm that samples spanning trees efficiently using Gaussian elimination and approximate effective resistances, surpassing prior determinant and walk-based techniques.

Findings

Achieves $ ilde{O}(n^{4/3}m^{1/2}+n^{2})$ runtime for weighted graphs.

Improves unweighted graph sampling time to $ ilde{O}( ext{min}igigrace{n^{ ext{omega}},m ext{sqrt}(n),m^{4/3}igigrace}$.

Introduces an efficient method to compute approximate effective resistances without Johnson-Lindenstrauss.

Abstract

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in $\tilde{O}(n^{4/3}m^{1/2}+n^{2})$ time (The $\tilde{O}(\cdot)$ notation hides $\operatorname{polylog}(n)$ factors). 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, $O(n^\omega)$. For the special case of unweighted graphs, this improves upon the best previously known running time of $\tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\})$ for $m \gg n^{5/3}$ (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute $\epsilon$-approximate effective resistances for a set $S$ of vertex pairs via approximate Schur complements in $\tilde{O}(m+(n + |S|)\epsilon^{-2})$ time, without using the Johnson-Lindenstrauss lemma which requires $\tilde{O}( \min\{(m + |S|)\epsilon^{-2}, m+n\epsilon^{-4} +|S|\epsilon^{-2}\})$ time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate.