A Tight Upper Bound on the Probabilistic Embedding of Series-Parallel Graphs

Abstract

We prove that every unweighted series-parallel graph can be probabilistically embedded into its spanning trees with logarithmic distortion. This is tight due to an $\Omega(\log n)$ lower bound established by Gupta, Newman, Rabinovich, and Sinclair on the distortion required to probabilistically embed the n-vertex diamond graph into a collection of dominating trees. Our upper bound is gained by presenting a polynomial time probabilistic algorithm that constructs spanning trees with low expected stretch. This probabilistic algorithm can be derandomized to yield a deterministic polynomial time algorithm for constructing a spanning tree of a given (unweighted) series-parallel graph G, whose communication cost is at most $O(\log n)$ times larger than that of G.