TY - JOUR

T1 - Terminal embeddings

AU - Elkin, Michael

AU - Filtser, Arnold

AU - Neiman, Ofer

N1 - Funding Information:
Supported in part by ISF grant No. (1817/17), and by BSF grant No. 2015813.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2017/10/12

Y1 - 2017/10/12

N2 - 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⊆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˜(logr)-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].

AB - 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⊆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˜(logr)-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].

KW - Distortion

KW - Embedding

KW - Terminals

UR - http://www.scopus.com/inward/record.url?scp=85026653060&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2017.06.021

DO - 10.1016/j.tcs.2017.06.021

M3 - Article

AN - SCOPUS:85026653060

VL - 697

SP - 1

EP - 36

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -