{"title":"Local Global Tradeoffs in Metric Embeddings","authors":"M. Charikar, K. Makarychev, Yury Makarychev","doi":"10.1137/070712080","DOIUrl":null,"url":null,"abstract":"Suppose that every k points in a metric space X are D-distortion embeddable into lscr 1. We give upper and lower bounds on the distortion required to embed the entire space X into lscr 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 lscr 1 with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"41","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/070712080","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 41
Abstract
Suppose that every k points in a metric space X are D-distortion embeddable into lscr 1. We give upper and lower bounds on the distortion required to embed the entire space X into lscr 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 lscr 1 with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.