In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.

The main question studied in this article may be viewed as a non-linear analog of Dvoretzky's Theorem in Banach space theory or as part of Ramsey Theory in combinatorics.

This article deals only with what might be called the geometrization of combinatorics. Namely, the idea that viewing combinatorial objects from a geometric perspective often yields unexpected insights. Even more concretely, we concentrate on finite metric spaces and their embeddings

INTRODUCTION An n-point metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their dissimilarity (computed, say, by comparing their DNA). It is dicult to see any structure in a large table of numbers, and so we would like to represent a given metric space in a more comprehensible way. For example, it would be very nice if we could assign to each x 2 X a point f(x) in the plane in such a way that D(x; y) equals the Euclidean distance of f(x) and f(y). Such a representation would allow us to see the structure of the metric space: tight clusters, isolated points, and so on. Another advantage would be that the metric would now be represented by only 2n real numbers, the coordinates of the n points in the plane, instead of numbers as before. Moreover, many quantities concern

The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a well-shaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other

We give a new lower bound for the rectilinear crossing number cr(n) of the complete geometric graph Kn. Weprovethatcr(n) ≥ 1 ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦ n n−1 n−2 n−3 andweextendtheproofof

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The radius of a k-dimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The k-at radius of P , which we denote by R k (P ), is the minimum, over all k-dimensional ats F , of RD(F ; P ). We consider the problem of computing R k (P ) for a given set of points P .

A common approach for solving computational problems over a difficult metric space is to embed the “hard ” metric into L1, which admits efficient algorithms and is thus considered an “easy ” metric. This approach has proved successful or partially successful for important spaces such as the edit distance, but it also has inherent limitations: it is provably impossible to go below certain approximation for some metrics. We propose a new approach, of embedding the difficult space into richer host spaces, namely iterated products of standard spaces like ℓ1 and ℓ∞. We show that this class is rich since it contains useful metric spaces with only a constant distortion, and, at the same time, it is tractable and admits efficient algorithms. Using this approach, we obtain for example the first nearest neighbor data structure with O(log log d) approximation for edit distance in nonrepetitive strings (the Ulam metric). This approximation is exponentially better than the lower bound for embedding into L1. Furthermore, we give constant factor approximation for two other computational problems. Along the way, we answer positively a question posed in [Ajtai, Jayram, Kumar, and Sivakumar, STOC 2002]. One of our algorithms has already found applications for smoothed edit distance over 0-1 strings [Andoni and Krauthgamer, ICALP 2008]. 1