Terminal embeddings

In this paper we study terminal embeddings, in which one is given a finite metric (X, dx) (or a graph G = (V, E)) and a subset K subset of X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve approximate to vertical bar K vertical bar center dot vertical bar X vertical bar pairs, the distortion depends only on vertical bar K vertical bar, rather than on vertical bar X vertical bar. We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X x X and with respect to K x X. Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [10] devised an O (root logr)-approximation algorithm for sparsest cut instances with r demands. Building on their framework, we provide an O (root log vertical bar K vertical bar)-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since vertical bar K vertical bar <= r, our bound generalizes that of [10]. (C) 2017 Elsevier B.V. All rights reserved.