Ittai Abraham

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

Ittai Abraham

Yair Bartal

Ofer Neiman

 

         Abstract

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 ε, with the guarantee that for each ε the distortion of a fraction 1-ε of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and lq-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/√ε). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O(1/√ε). These bounds are tight even for embedding in arbitrary trees.
For probabilistic embedding into spanning trees we prove a scaling distortion of ~O(log2 (1/ε)), which implies constant lq-distortion for every fixed q<∞.

 

[Extended TR version  pdf]