When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.242
URN: urn:nbn:de:0030-drops-53064
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5306/
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### Terminal Embeddings

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### Abstract

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 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 approx |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 \times X and with respect to K * X. Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, Arora et. al. devised an ~O(sqrt(log(r))-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an ~O(sqrt(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 Arora et al.

### BibTeX - Entry

@InProceedings{elkin_et_al:LIPIcs:2015:5306,
author =	{Michael Elkin and Arnold Filtser and Ofer Neiman},
title =	{{Terminal Embeddings}},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
pages =	{242--264},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-89-7},
ISSN =	{1868-8969},
year =	{2015},
volume =	{40},
editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},