TY - GEN

T1 - Terminal embeddings

AU - Elkin, Michael

AU - Filtser, Arnold

AU - Neiman, Ofer

N1 - Publisher Copyright:
© Michael Elkin, Arnold Filtser, and Ofer Neiman.

PY - 2015/8/1

Y1 - 2015/8/1

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, [7] devised an Õ(√log r)-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an Õ(√ 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 [7].

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, [7] devised an Õ(√log r)-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an Õ(√ 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 [7].

KW - Distortion

KW - Embedding

KW - Terminals

UR - http://www.scopus.com/inward/record.url?scp=84958549366&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2015.242

DO - 10.4230/LIPIcs.APPROX-RANDOM.2015.242

M3 - Conference contribution

AN - SCOPUS:84958549366

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 242

EP - 264

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015

A2 - Garg, Naveen

A2 - Jansen, Klaus

A2 - Rao, Anup

A2 - Rolim, Jose D. P.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015

Y2 - 24 August 2015 through 26 August 2015

ER -