Terminal embeddings

In this paper we study terminal embeddings, in which one is given a finite metric (X,d_X) (or a graph G=(V,E)) and a subset K⊆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 ≈|K|⋅|X| pairs, the distortion depends only on |K|, rather than on |X|. 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 and with respect to K×X. Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [10] devised an O˜(log⁡r)-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an O˜(log⁡|K|)- approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K|≤r, our bound generalizes that of [10].