Sampling Random Spanning Trees Faster Than Matrix Multiplication

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in (O) over tilde (n(5/3)m(1/3)) 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(n(omega)). For the special case of unweighted graphs, this improves upon the best previously known running time of (O) over tilde (min{n(omega), m root n, m(4/3)}) form m >> n(7/4) (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 (O) over tilde (m + (n + vertical bar S vertical bar)is an element of(-2)) time, without using the Johnson-Lindenstrauss lemma which requires (O) over tilde (min {(m +vertical bar S vertical bar)is an element of(-2), m + n is an element of(-4) + vertical bar S vertical bar is an element of(-2)}) time. We combine this approximation procedure with an error correction procedure for handling edges where our estimate isn't suffciently accurate.