Terminal embeddings

Michael Elkin, Arnold Filtser, Ofer Neiman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

Abstract

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].

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015
EditorsNaveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages242-264
Number of pages23
ISBN (Electronic)9783939897897
DOIs
StatePublished - 1 Aug 2015
Externally publishedYes
Event18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United States
Duration: 24 Aug 201526 Aug 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume40
ISSN (Print)1868-8969

Conference

Conference18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015
Country/TerritoryUnited States
CityPrinceton
Period24/08/1526/08/15

Bibliographical note

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

Funding

FundersFunder number
Seventh Framework Programme303809

    Keywords

    • Distortion
    • Embedding
    • Terminals

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