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101
Bounded geometries, fractals, and lowdistortion embeddings
"... 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 ..."
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Cited by 154 (31 self)
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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 (lowdimensional) 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.
Meridian: A Lightweight Network Location Service without Virtual Coordinates
 In SIGCOMM
, 2005
"... 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 multiresolution rings, query routing with direct measurements, and gossip protocols for diss ..."
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Cited by 139 (7 self)
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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 multiresolution 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 largescale 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 lowdimensional Euclidean metric, or a metric of low doubling dimension. Large scale simulations, based on latency measurements from 6.25 million nodepairs as well as an implementation deployed on PlanetLab show that the framework is accurate and effective.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Distance Estimation and Object Location via Rings of Neighbors
 In 24 th Annual ACM Symposium on Principles of Distributed Computing (PODC
, 2005
"... We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased distance estimation [33]. Fo ..."
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Cited by 64 (4 self)
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We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased 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
On Hierarchical Routing in Doubling Metrics
, 2005
"... We study the problem of routing in doubling metrics, and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with small amount of routing information stored at each vertex). We say that a metric (X, d) has doubling dimension dim(X) at most α ..."
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Cited by 58 (8 self)
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We study the problem of routing in doubling metrics, and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with small amount of routing information stored at each vertex). We say that a metric (X, d) has doubling dimension dim(X) at most α if every set of diameter D can be covered by 2 α sets of diameter D/2. (A doubling metric is one whose doubling dimension dim(X) is a constant.) We show how to perform (1 + τ)stretch routing on metrics for any 0 < τ ≤ 1 with routing tables of size at most (α/τ) O(α) log 2 ∆ bits with only (α/τ) O(α) log ∆ entries, where ∆ is the diameter of the graph; hence the number of routing table entries is just τ −O(1) log ∆ for doubling metrics. These results extend and improve on those of Talwar (2004). We also give better constructions of sparse spanners for doubling metrics than those obtained from the routing tables above; for τ> 0, we give algorithms to construct (1 + τ)stretch spanners for a metric (X, d) with maximum degree at most (2 + 1/τ) O(dim(X)) , matching the results of Das et al. for Euclidean metrics.
Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces
"... A metric space X has Markov type 2, if for any reversible finitestate 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 d ..."
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Cited by 41 (24 self)
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A metric space X has Markov type 2, if for any reversible finitestate 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 biLipschitz 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
LocalitySensitive Binary Codes from ShiftInvariant Kernels,” Advances in neural information processing systems
, 2009
"... This paper addresses the problem of designing binary codes for highdimensional data such that vectors that are similar in the original space map to similar binary strings. We introduce a simple distributionfree encoding scheme based on random projections, such that the expected Hamming distance be ..."
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Cited by 40 (1 self)
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This paper addresses the problem of designing binary codes for highdimensional data such that vectors that are similar in the original space map to similar binary strings. We introduce a simple distributionfree encoding scheme based on random projections, such that the expected Hamming distance between the binary codes of two vectors is related to the value of a shiftinvariant kernel (e.g., a Gaussian kernel) between the vectors. We present a full theoretical analysis of the convergence properties of the proposed scheme, and report favorable experimental performance as compared to a recent stateoftheart method, spectral hashing. 1
Measure Theoretic Laws for lim sup Sets
, 2004
"... Given a compact metric space (Ω, d) equipped with a nonatomic, probability measure m and a positive decreasing function ψ, we consider a natural class of limsup subsets Λ(ψ) of Ω. The classical limsup set W(ψ) of ‘ψ–approximable ’ numbers in the theory of metric Diophantine approximation fall withi ..."
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Cited by 32 (8 self)
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Given a compact metric space (Ω, d) equipped with a nonatomic, probability measure m and a positive decreasing function ψ, we consider a natural class of limsup subsets Λ(ψ) of Ω. The classical limsup set W(ψ) of ‘ψ–approximable ’ numbers in the theory of metric Diophantine approximation fall within this class. We establish sufficient conditions (which are also necessary under some natural assumptions) for the m–measure of Λ(ψ) to be either positive or full in Ω and for the Hausdorff fmeasure to be infinite. The classical theorems of KhintchineGroshev and Jarník concerning W(ψ) fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantine approximation including those for real, complex and padic fields associated with both independent and dependent quantities. Applications also include those to Kleinian groups and rational maps. Compared to earlier works our framework allows us to successfully remove many unnecessary conditions and strengthen results such as Jarník’s theorem and the BakerSchmidt theorem. In particular, the strengthening of Jarník’s theorem opens up the DuffinSchaeffer conjecture for Hausdorff measures.
Nagata dimension, quasisymmetric embeddings and Lipschitz extensions, arXive:math
, 2004
"... The largescale geometry of metric spaces has been the subject of intense investigation in recent years. In his essay on asymptotic invariants of infinite ..."
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Cited by 28 (1 self)
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The largescale geometry of metric spaces has been the subject of intense investigation in recent years. In his essay on asymptotic invariants of infinite