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

This paper addresses the basic question of how well can a tree 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 ∈, with the guarantee that for each ∈ the distortion of a fraction 1 - ∈ of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and ℓ_q-distortions are small. Specifically, our embeddings have constant average distortion and O(√log n)ℓ₂-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O(√1/∈). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O(√1/∈). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of Õ (log²(1/∈)), which implies constant ℓ_q-distortion for every fixed q < ∞.