License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5306/
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Elkin, Michael ; Filtser, Arnold ; Neiman, Ofer

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},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5306},
  URN =		{urn:nbn:de:0030-drops-53064},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.242},
  annote =	{Keywords: embedding, distortion, terminals}
}

Keywords: embedding, distortion, terminals
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 13.08.2015

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