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 q-distortions 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( p1/ ). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O( p1/ ). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into...

• Type: Conference
• Year: 2007
• Where: SODA
• Authors: Ittai Abraham, Yair Bartal, Ofer Neiman
• Keywords: Algorithms, Average Distortion, Distortion, Scaling Distortion, SODA 2007