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arXiv · 0908.1448

Faster generation of random spanning trees

Jonathan A. Kelner·Aleksander Madry
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Abstract

In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative $(1+δ)$ of uniform in expected time $\TO(m\sqrt{n}\log 1/δ)$. This improves the sparse graph case of the best previously known worst-case bound of $O(\min \{mn, n^{2.376}\})$, which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks, and we use this to introduce a new approach to the problem that integrates discrete random walk-based techniques with continuous linear algebraic methods. We believe that our use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is a useful paradigm that will find further applications in algorithmic graph theory.

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Jonathan A. Kelner, Aleksander Madry. 2009-08-11. Faster generation of random spanning trees. https://arxiv.org/abs/0908.1448

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