Metric Embedding via Shortest Path Decompositions

Abstract

We study the problem of embedding weighted graphs of pathwidth k into lp spaces. Our main result is an O(k^min{1/p,1/2})-distortion embedding. For p = 1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p > 1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p > 2 our embedding also implies improved distortion on bounded treewidth graphs (O((k logn)^{1/p})). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.