Random walks in distributed networks are useful primitives with numerous applications. In this paper, we focus on efficient distributed algorithms for performing random walks and their application to generating random spanning trees (RST) in arbitrary networks. The goal is to minimize the number of rounds required to output the positions of vertices in a random walk starting from a given source and to generate an RST in the standard CONGEST model.

We present the first sublinear-in-n round algorithm for sampling an approximately uniform spanning tree of an n-vertex graph in the CongestedClique model of distributed computing. In particular, our algorithm requires Õ (n0.657) rounds for sampling a spanning tree within total variation distance 1/nc, for arbitrary constant c > 0, from the uniform distribution. More precisely, our algorithm requires Õ (n1/2+α) rounds, where O (nα) is the running time of matrix multiplication in the CongestedClique model (currently α = 1-2/ω = 0.157, where ω is the sequential matrix multiplication time exponent). We can adapt our algorithm to give exact rather than approximate samples, but with a larger, though still o (n), runtime of Õ(n2/3+α) = O(n0.824).