The doubling constant of a metric space (X; d) is thesmallest value * such that every ball in X can be covered by * balls of half the radius. The doubling dimension of X isthen defined as dim(X) = log2 *. A metric (or sequence ofmetrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaceswhich contains many families of metrics that occur in applied settings.We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider bothgeneral doubling metrics, as well as more restricted families such as those arising from trees, from graphs excludinga fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, andan analysis of a fractal in the plane due to Laakso [21]. Finally, we discuss some applications and point out a centralopen question regarding dimensionality reduction in L2.

Bi-Lipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every

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. 1

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.

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We define a natural notion of efficiency for approximate nearest-neighbor (ANN) search in general n-point metric spaces, namely the existence of a randomized algorithm which answers (1 + ε)-approximate nearest neighbor queries in polylog(n) time using only polynomial space. We then study which families of metric spaces admit efficient ANN schemes in the black-box model, where only oracle access to the distance function is given, and any query consistent with the triangle inequality may be asked. For ε < 2 5, we offer a complete answer to this problem. Using the notion of metric dimension defined in [GKL03] (à la [Ass83]), we show that a metric space X admits an efficient (1+ε)-ANN scheme for any ε < 2 5 if and only if dim(X) = O(log log n). For coarser approximations, clearly the upper bound continues to hold, but there is a threshold at which our lower bound breaks down—this is precisely when points in the “ambient space ” may begin to affect the complexity of “hard ” subspaces S ⊆ X. Indeed, we give examples which show that dim(X) does not characterize the black-box complexity of ANN above the threshold. Our scheme for ANN in low-dimensional metric spaces is the first to yield efficient algorithms without relying on any additional assumptions on the input. In previous approaches (e.g., [Cla99, KR02, KL04, HKMR04]), even spaces with dim(X) = O(1) sometimes required Ω(n) query times. 1

A number of recent papers in the networking community study the distance matrix defined by the node-to-node latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space. In such algorithms it is feasible to measure distances among only a linear or near-linear number of node pairs; the rest of the distances are simply not available. Moreover, for applications it is desirable to spread the load evenly among the participating nodes. Indeed, several recent studies use this ’fully distributed ’ approach and achieve, empirically, a low distortion for all but a small fraction of node pairs. This is concurrent with the large body of theoretical work on metric embeddings, but there is a fundamental distinction: in the theoretical approaches to metric embeddings, full and centralized access to the distance matrix is assumed and heavily used. In this paper we present the first fully distributed embedding algorithm with provable distortion guarantees for doubling metrics (which have been proposed as a reasonable abstraction of Internet latencies), thus providing some insight into the empirical success of the recent Vivaldi algorithm [7]. The main ingredient of our embedding algorithm is an improved fully distributed algorithm for a more basic problem of triangulation, where the triangle inequality is used to infer the distances that have not been measured; this problem received a considerable attention in the networking community, and has also been studied theoretically in [19]. We use our techniques to extend ɛ-relaxed embeddings and triangulations to infinite metrics and arbitrary measures, and to improve on the approximate distance labeling scheme of Talwar [36]. 1

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, 14] 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. 1

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [29], we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into ℓ1 which exhibit

Abstract. We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [16]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of Z d ∞, the infinite graph with vertex set Z d and an edge (u, v) whenever ||u − v|| ∞ = 1. The growth rate of G, denoted ρG, is the minimum ρ such that every ball of radius r> 1 in G contains at most r ρ vertices. By simple volume arguments, dim(G) = Ω(ρG). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρG) for every graph G. Previously, it was unknown whether dim(G) could be bounded above by any function of ρG. We show that a weaker form of Levin’s conjecture holds by proving that dim(G) = O(ρG log ρG) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) = Ω(ρG log ρG). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρG). Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces posed by Linial [15] and independently by Benjamini and Schramm [22]. 1.