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Providence, Rhode Island

Oct. 21, 2007 to Oct. 23, 2007

ISBN: 0-7695-3010-9

pp: 713-723

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2007.64

ABSTRACT

Suppose that every k points in a metric space X are D-distortion embeddable into \ell _1. We give upper and lower bounds on the distortion required to embed the entire space X into \ell _1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into \ell _1 with distortion {\rm O}(D \times \log (\left| X \right|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For D = 1 + \delta we give a lower bound of \Omega (\log (\left| X \right|/k)/\log (1/\delta )); and for D = 1, we give a lower bound of \Omega (\log \left| X \right|/(\log k + \log \log \left| X \right|)). Our bounds significantly improve on the results of Arora, Lovész, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.

INDEX TERMS

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CITATION

Moses Charikar,
Konstantin Makarychev,
Yury Makarychev,
"Local Global Tradeoffs in Metric Embeddings",

*FOCS*, 2007, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2007, pp. 713-723, doi:10.1109/FOCS.2007.64