Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

Abstract

This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of $\epsilon$, with the guarantee that for each $\epsilon$ simultaneously, the distortion of a fraction $1-\epsilon$ of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/\epsilon})$. For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/\epsilon})$. These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the $\ell_2$ distortion is $O(\sqrt{\log n})$. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/\epsilon))$, which implies constant $\ell_q$-distortion for every fixed $q<\infty$.