How to get an exact sample from a generic Markov chain and sample a random spanning tree from a directed graph, both within the cover time

This paper shows how to obtain unbiased samples from an unknown Markov chain by observing it for O(T{sub c}) steps, where T{sub c} is the cover time. This algorithm improves on several previous algorithms, and there is a matching lower bound. Using the techniques from the sampling algorithm, we also show how to sample random directed spanning trees from a weighted directed graph, with arcs directed to a root, and probability proportional to the product of the edge weights. This tree sampling algorithm runs within 18 cover times of the associated random walk, and is more generally applicable than the algorithm of Broder and Aldous.

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