# Local Global Tradeoffs in Metric Embeddings

## Abstract

*k*points in a

*n*point 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 $O(D\times\log(n/k))$. 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 $\Omega(\log(n/k)/\log(1/\delta))$; and for $D=1$, we give a lower bound of $\Omega(\log n/(\log k+\log\log n))$. Our bounds significantly improve on the results of Arora et al. who initiated a study of these questions.

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#### History

**Submitted**: 2 January 2008

**Accepted**: 13 April 2009

**Published online**: 30 April 2010

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