We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms for the following problems: Approximate nearest neighbor search, well-separated pair decomposition, spanner construction, compact representation scheme, doubling measure, and computation of the (approximate) Lipschitz constant of a function. In all cases, the running (preprocessing) time is near-linear and the space being used is linear.This view seems to be shared by many recent algorithmic papers on doubling metrics, but it still awaits a convincing empirical and/or theoretical support.2. Even if one is only interested in questions on Euclidean point sets, it makes sense to strip the techniques being used to their bare essentials, obtaining better understanding of the problems and conceptually simpler solutions.More arguments along these lines can be found in [14], where the author advocates this approach. In general, it is impossible to directly apply algorithmic results developed for fixed dimensional Euclidean space to doubling metrics, since there exists doubling metrics that can not embedded in Hilbert space with low distortion of the distances [41,35]. Hence, some of the aforementioned works apply notions and techniques from fixed dimensional Computational Geometry and extend them to finite metric spaces.In particular, Talwar [44] showed that one can extend the notion of well-separated pairs decomposition (WSPD) of [11] to spaces with low doubling dimension. Specifically, he shows that for every set P of n points having doubling dimension dim, and every ε > 0, there exists WSPD, with separation 1/ε and O(nε −O(dim) log Φ) pairs, where dim is the doubling dimension of the finite metric space, and Φ is the spread of the point set, which is the ratio between the diameter of P and the distance between the closest pair of points in P . This is weaker than the result of Callahan and Kosaraju [11] for Euclidean space, which does not depend on the spread of the point set.Krauthgamer and Lee [34] showed a data structure for answering (1 + ε)-approximate nearest neighbor queries on point set P with spread Φ. Their data structure supports insertions in O(log Φ log log Φ) time. The preprocessing time is O(n log Φ log log Φ) (this is by inserting the points one by one), and the query time is O(log Φ + ε −O(dim) ). In IR d for fixed d, one can answer such queries in O(log log(Φ/ε)) time, using near linear space, see [24] and references therein (in fact, it is possible to achieve constant query time using slightly larger storage [26]). Note however, that the latter results strongly use the Euclidean structure. Recently, Krauthgamer and Lee [33] overcame the restriction on the spread, presenting a data-structure with nearly quadratic space, and logarithmic query time.Underlining all those results, is the notion of hierarchical nets. Intuitively, hierarchical nets are sequences of larger and larger subsets of the underlining s...

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L p coarsely or uniformly embeds into L q . We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matoušek.

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0, 1), every n-point metric space has a subset of size n 1−ε which embeds into Hilbert space with distortion O(1/ε). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [32]. Namely, we show that for every n-point metric space X, and k ≥ 1, there exists an O(k)-approximate distance oracle whose storage requirement is O(n 1+1/k), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

No abstract

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

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 Fréchet 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( √ α X · log n), where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O( (log λ X ) log n) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ 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 ≤ k ≤ 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 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 ).

This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class.Our main theorem states that for any > 0, every n point metric space contains a subspace of size at least n 1− which is embeddable in an ultrametric with O( log(1/ ) ) distortion. This in particular provides a bound for embedding in Euclidean spaces. The bound on the distortion is tight up to the log(1/ ) factor even for embedding in arbitrary Euclidean spaces. This result can be viewed as a non-linear analog of Dvoretzky's theorem, a cornerstone of modern Banach space theory and convex geometry.Our main Ramsey-type theorem and techniques naturally extend to give theorems for classes of hierarchically wellseparated trees which have algorithmic implications, and can be viewed as the solution of a natural clustering problem.We further include a comprehensive study of various other aspects of the metric Ramsey problem. The full version appears mostly in a manuscript entitled the same. Some parts appear in the papers entitled "On some low distortion metric Ramsey problems", "On Fréchet embedding of metric spaces", and "On Lipschitz embedding of ultrametrics in low dimension" by the same authors. The manuscripts can be downloaded from

We study the metric properties of finite subsets of L1. 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 L1.We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O( √ log n), yielding the first evidence that the conjectured worst-case bound of O( √ log n) is valid. We also address the issue of dimension reduction in Lp for p ∈ (1, 2). We resolve a question left open in [4] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [3,16] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space. IntroductionThis paper is devoted to the analysis of metric properties of finite subsets of L 1 . Such metrics occur in many important algorithmic contexts, and their analysis is key to progress on some fundamental problems. For instance, an O(log n)-approximate max-flow/min-cut theorem proved elusive for many years until, in [18,2], it was shown to follow from a theorem of Bourgain stating that every metric on n points embeds into L 1 with distortion O(log n).The importance of L 1 metrics has given rise to many problems and conjectures that have attracted a lot of attention in recent years. to Four basic problems of this type are as follows . (We recall that a squared-ℓ 2 metric is a space (X, d) for which (X, d 1/2 ) embeds isometrically in a Hilbert space.) Each of these questions has been asked many times before; we refer to [21,22,17,11], in particular. Despite an immense amount of interest and effort, the metric properties of L 1 have proved quite elusive; *