Perfect Sampling of Random Spanning Trees
نویسنده
چکیده
Let G be a weighted directed graph where the edge weights are non-negative real numbers. An arborescence or (directed) spanning tree with root r is a connected subgraph of G such that there is a unique directed path from v to r for each v ∈ V (G). The weight of a spanning tree T will be the product of the weights of the edges of T . Let Υ(G) be the probability distribution on the set of all spanning trees of G where the probability of a tree is proportional to its weight, and let Υr (G) be the probability distribution of the set of all spanning trees of G with root r where the probability of a tree with root r is proportional to its weight. We are interested in efficient algorithms for generating random samples from Υ(G) and Υr (G). The algorithms we will be discussing here are based on random walks (more generally Markov chains.) We say that G is stochastic if for each vertex the sum of the weights of all outgoing arcs is equal to 1. If G is stochastic then we can of course think of it as a Markov chain. Otherwise we can define two related stochastic graphs G and G̃. G is obtained by normalizing the weights of the outgoing arcs for each vertex so that their sum is 1. G̃ is obtained by first adding a weighted loop to each vertex so that their out-degrees are all equal and then normalizing the weights of all outgoing arcs so that the out-degree of each vertex is 1. It is easy to check that Υ(G) = Υ(G̃) and Υr (G) = Υr (G) and therefore we will use the Markov chain associated with G̃ when sampling from Υ(G), and use the Markov chain associated with G when sampling from Υr (G). (Note: we also have Υr (G̃) = Υr (G) but it is preferable to use G when sampling from Υr (G) since it produces more efficient algorithms due to smaller cover and mean hitting times.)
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تاریخ انتشار 2005