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Providence, Rhode Island
Oct. 21, 2007 to Oct. 23, 2007
ISBN: 0-7695-3010-9
pp: 713-723
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.
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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
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