Metric structures in L_1: Dimension, snowflakes, and average distortion

We study the metric properties of finite subsets of L_1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L_1.Comment: 9 pages, 1 figure. To appear in European Journal of Combinatorics. Preliminary version appeared in LATIN '0

On the Hausdorff dimension of ultrametric subsets in R^n

For every e>0, any subset of R^n with Hausdorff dimension larger than (1-e)n must have ultrametric distortion larger than 1/(4e).Comment: 4 pages, improved layou

Euclidean quotients of finite metric spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings into l_p, and the particular case of the hypercube.Comment: 36 pages, 0 figures. To appear in Advances in Mathematic

Measured descent: A new embedding method for finite metrics

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where alpha_X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding, where \lambda_X is the doubling constant of X. Since \lambda_X\le n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, ``low-dimensional,'' improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 \leq k \leq n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)^2).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric & Functional Analysis. This version fixes a subtle error in Section 2.

Finite Volume Spaces and Sparsification

We introduce and study finite $d$-volumes - the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define $\ell_1$-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any $d$-volume with $O(n^d)$ multiplicative distortion. On the other hand, contrary to Bourgain's theorem for $d=1$, there exists a $2$-volume that on $n$ vertices that cannot be approximated by any $\ell_1$-volume with distortion smaller than $\tilde{\Omega}(n^{1/5})$. We further address the problem of $\ell_1$-dimension reduction in the context of $\ell_1$ volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any $\ell_1$ metric on $n$ points can be $(1+ \epsilon)$-approximated by a sum of $O(n/\epsilon^2)$ cut metrics, improving over the best previously known bound of $O(n \log n)$ due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of graph Sparsification, and develop general methods with a wide range of applications.Comment: previous revision was the wrong file: the new revision: changed (extended considerably) the treatment of finite volumes (see revised abstract). Inserted new applications for the sparsification technique