Metric Embedding via Shortest Path Decompositions

We study the problem of embedding shortest-path metrics of weighted graphs intoℓp spaces. We introduce a new embedding technique based on low-depth decompositions of a graphvia shortest paths. The notion of shortest path decomposition (SPD) depth is inductively defined:A (weighed) path graph has SPD depth 1. General graph has an SPD of depth k if it contains ashortest path whose deletion leads to a graph, each of whose components has SPD depth at mostk − 1. In this paper we give an O(kmin{1/p,1/2})-distortion embedding for graphs of SPD depth atmost k. This result is asymptotically tight for any fixed p > 1, while for p = 1 it is tight up to secondorder terms. As a corollary of this result, we show that graphs having pathwidth k embed into ℓpwith distortion O(kmin{1/p,1/2}). For p = 1, this improves over the best previous bound of Lee andSidiropoulos that was exponential in k; moreover, for other values of p it gives the first embeddingswhose distortion is independent of the graph size n. Furthermore, we use the fact that planar graphshave SPD depth O(log n) to give a new proof that any planar graph embeds into ℓ1 with distortionO(√log n). Our approach also gives new results for graphs with bounded treewidth, and for graphsexcluding a fixed minor.