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arXiv · 1501.00267

Fast Generation of Random Spanning Trees and the Effective Resistance Metric

Aleksander Madry·Damian Straszak·Jakub Tarnawski
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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.

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Aleksander Madry, Damian Straszak, Jakub Tarnawski. 2015-01-01. Fast Generation of Random Spanning Trees and the Effective Resistance Metric. https://doi.org/10.1137/1.9781611973730.134

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