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 qdistortions are small. Specifically, our embeddings have constant average distortion and O( √ log n) 2 -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 ofÕ(log 2 (1/ )), which implies constant qdistortion for every fixed q < ∞.
Recently REF give an embedding from a weighted graph into one of its spanning trees, whose average distortion over all pairs is constant.
Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion
SODA '07
Mathematics Computer Science