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2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2007)
Providence, Rhode Island
Oct. 21, 2007 to Oct. 23, 2007
ISSN: 0272-5428
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, Yury Makarychev, Konstantin Makarychev, "Local Global Tradeoffs in Metric Embeddings", 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, vol. 00, no. , pp. 713-723, 2007, doi:10.1109/FOCS.2007.64
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