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.

This paper introduces a lightweight, scalable and accurate framework, called Meridian, for performing node selection based on network location. The framework consists of an overlay network structured around multi-resolution rings, query routing with direct measurements, and gossip protocols for dissemination. We show how this framework can be used to address three commonly encountered problems, namely, closest node discovery, central leader election, and locating nodes that satisfy target latency constraints in large-scale distributed systems without having to compute absolute coordinates. We show analytically that the framework is scalable with logarithmic convergence when Internet latencies are modeled as a growthconstrained metric, a low-dimensional Euclidean metric, or a metric of low doubling dimension. Large scale simulations, based on latency measurements from 6.25 million node-pairs as well as an implementation deployed on PlanetLab show that the framework is accurate and effective.

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.

We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: low-stretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulation-based distance estimation [33]. Focusing on metrics of low doubling dimension, we approach these problems with a common technique called rings of neighbors, which refers to a sparse distributed data structure that underlies all our constructions. Apart from improving the previously known bounds for these problems, our contributions include extending Kleinberg’s small world model to doubling metrics, and a short proof of the main result in Chan et al. [14]. Doubling dimension is a notion of dimensionality for general metrics that has recently become a useful algorithmic concept in the theoretical computer science literature. 1

The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized. We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks.

. It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any ffi -hyperbolic metric space embeds isometrically into a complete geodesic ffi -hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. 1. Introduction The study of Gromov hyperbolic spaces has been largely motivated and dominated by questions about Gromov hyperbolic groups. This paper studies the geometry of Gromov hyperbolic spaces without reference to any group or group action. One of our main theorems is 1.1. Embedding Theorem. Let X be a Gromov hyperbolic geodesic metric spa...

A metric space X has Markov type 2, if for any reversible finite-state Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its bi-Lipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type 2 (in particular, Lp for p> 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp. 1

The distributed complexity of computing a maximal independent set in a graph is of both practical and theoretical importance. While there exists an elegant O(log n) time randomized algorithm for general graphs [20], no deterministic polylogarithmic algorithm is known. In this paper, we study the problem in graphs with bounded growth, an important family of graphs which includes the well-known unit disk graph and many variants thereof. Particularly, we propose a deterministic algorithm that computes a maximal independent set in time O(log \Delta * log*n) in graphs with bounded growth, where n and \Delta denote the number of nodes and the maximal degree in G, respectively.

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