Near-Optimal Distributed Tree Embedding

Tree embeddings are a powerful tool in the area of graph approximation algorithms. Roughly speaking, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms n-node metrics into (probability distributions over) trees, while stretching each pairwise distance by at most anO(log n) factor in expectation. This O(log n) stretch is optimal. Khan et al. [PODC’08] present a distributed algorithm that implements FRT in O(SPD log n) rounds, where SPD is the shortest-path-diameter of the weighted graph, and they explain how to use this embedding for various distributed approximation problems. Note that SPD can be as large as Θ(n), even in graphs where the hop-diameter D is a constant. Khan et al. noted that it would be interesting to improve this complexity. We show that this is indeed possible. More precisely, we present a distributed algorithm that constructs a tree embedding that is essentially as good as FRT in Õ(min{n,SPD} + D) rounds, for any constant ε > 0. A lower bound of Ω̃(min{n,SPD}+D) rounds follows from Das Sarma et al. [STOC’11], rendering our round complexity near-optimal.