On average distortion of embedding metrics into the line and into L 1

Abstract

We introduce and study the notion of the average distortion of a nonexpanding embedding of one metric space into another. Less sensitive than the multiplicative metric distortion, the average distortion captures well the global picture, and, overall, is a quite interesting new measure of metric proximity, related to the concentration of measure phenomenon. We establish close mutual relations between the MinCut- MaxFlow gap in a uniform-demand multicommodity flow, and the average distortion of embedding the suitable (dual) metric into l1. These relations are exploited to show that the shortest-path metrics of special (e.g., planar, bounded treewidth, etc.) graphs embed into l1 with constant average distortion. The main result of the paper claims that this remains true even if l1 is replaced with the line. This result is further sharpened for graphs of a bounded treewidth.