Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion

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 ε, with the guarantee that for each ε simultaneously, the distortion of a fraction 1 - ε of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion O(formula presented). For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion O(formula presented). 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 ℓ₂ distortion is O(formula presented). For probabilistic embedding into spanning trees we prove a scaling distortion of Õ(log²(1/ε)), which implies constant ℓ_q-distortion for every fixed q < ∞.