A tight upper bound on the probabilistic embedding of series-parallel graphs

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 Ω(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.