Distributed MST and Routing in Almost Mixing Time

Abstract

We present a randomized distributed algorithm that computes a minimum spanning tree in τ(G) · 2O(√(log n log log n))) rounds, in any n-node graph G with mixing time τ(G). This result provides a sub-polynomial complexity for a wide range of graphs of practical interest, and goes below the celebrated Ω(D+ √n) lower bound of Das Sarma et al. [STOC'11] which holds for some worst-case general graphs. The core novelty in this result is a distributed method for permutation routing. In this problem, one is given a number of source-destination pairs, and we should deliver one packet from each source to its destination, all in parallel, in the shortest span of time possible. Our algorithm allows us to route and deliver all these packets in τ(G) · 2O(√(log n log log n)) rounds, assuming that each node v is the source or destination for at most dG(v) packets. The main technical ingredient in this routing result is a certain hierarchical embedding of good-expansion random graphs on the base graph, which we believe can be of interest well beyond this work.