On average distortion of embedding metrics into the line and into L₁

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.