NDLI: Sampling random spanning trees faster than matrix multiplication

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Sampling random spanning trees faster than matrix multiplication

Content Provider	ACM Digital Library
Author	Durfee, David Sachdeva, Sushant Rao, Anup B. Kyng, Rasmus Peebles, John
Abstract	We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in (n5/3 $}m^{1/3})$ 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(nÏ ). For the special case of unweighted graphs, this improves upon the best previously known running time of Ã (min{nÏ ,mâ $n,m^{4/3}\})$ for m â « $n^{7/4}$ (Colbourn et al. â 96, Kelner-Madry â 09, Madry et al. â 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 -approximate effective resistances for a set S of vertex pairs via approximate Schur complements in (m+(n + \|S\|)â 2) time, without using the Johnson-Lindenstrauss lemma which requires ( min{(m + \|S\|)â 2, m+nâ 4 +\|S\|â 2}) time. We combine this approximation procedure with an error correction procedure for handling edges where our estimate isnâ t sufficiently accurate. .
Starting Page	730
Ending Page	742
Page Count	13
File Format	PDF
ISBN	9781450345286
DOI	10.1145/3055399.3055499
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-06-19
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Random spanning trees Approximate schur complement Graph sparsification Sampling algorithm
Content Type	Text
Resource Type	Article

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