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27
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 198 (40 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 190 (8 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.
Triangulation and Embedding using Small Sets of Beacons
, 2008
"... Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance ..."
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Cited by 96 (11 self)
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Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into lowdimensional Euclidean space. There is a fundamental distinction, however, between the theoretical approaches to the embedding problem and this recent Internetrelated work: in addition to computational limitations, Internet measurement algorithms operate under the constraint that it is only feasible to measure distances for a linear (or nearlinear) number of node pairs, and typically in a highly structured way. Indeed, the most common framework for Internet measurements of this type is a beaconbased approach: one chooses uniformly at random a constant number of nodes (‘beacons’) in the network, each node measures its distance to all beacons, and one then has access to only these measurements for the remainder of the algorithm. Moreover, beaconbased algorithms are often designed not for embedding but for the more basic problem of triangulation, in which one uses the triangle inequality to infer the distances that have not been measured. Here we give algorithms with provable performance guarantees for beaconbased triangulation and
On Hierarchical Routing in Doubling Metrics
"... 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 f ..."
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Cited by 67 (7 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 ff if every set of diameter D can be covered by 2ff sets of diameter D=2. (A doubling metric is one whose doubling dimension dim(X) is a constant.) We show how to perform (1 + o /)stretch routing on metrics for any 0! o / ^ 1 with routing tables of size at most (ff=o /)O(ff) log2 \Delta bits with only (ff=o /)O(ff) log \Delta entries, where \Delta is the diameter of the graph; hence the number of routing table entries is just o / \Gamma O(1) log \Delta 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 o / ? 0, we give algorithms to construct (1 + o /)stretch spanners for a metric (X; d) with maximum degree at most (2 + 1=o /)O(dim(X)), matching the results of Das et al.for Euclidean metrics.
A tractable approach to finding closest truncatedcommutetime neighbors in large graphs
 In Proc. UAI
, 2007
"... Recently there has been much interest in graphbased learning, with applications in collaborative filtering for recommender networks, link prediction for social networks and fraud detection. These networks can consist of millions of entities, and so it is very important to develop highly efficient t ..."
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Cited by 43 (7 self)
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Recently there has been much interest in graphbased learning, with applications in collaborative filtering for recommender networks, link prediction for social networks and fraud detection. These networks can consist of millions of entities, and so it is very important to develop highly efficient techniques. We are especially interested in accelerating random walk approaches to compute some very interesting proximity measures of these kinds of graphs. These measures have been shown to do well empirically (LibenNowell & Kleinberg, 2003; Brand, 2005). We introduce a truncated variation on a wellknown measure, namely commute times arising from random walks on graphs. We present a very novel algorithm to compute all interesting pairs of approximate nearest neighbors in truncated commute times, without computing it between all pairs. We show results on both simulated and real graphs of size up to 100, 000 entities, which indicate nearlinear scaling in computation time. 1
Faulttolerant spanners: Better and simpler
 In PODC
, 2011
"... A natural requirement for many distributed structures is faulttolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to verte ..."
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Cited by 14 (3 self)
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A natural requirement for many distributed structures is faulttolerance: after some failures in the underlying network, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults r for all stretch bounds. For stretch k ≥ 3 we design a simple transformation that converts every kspanner construction with at most f(n) edges into an rfaulttolerant kspanner construction with at most O(r 3 log n) · f(2n/r) edges. Applying this to standard greedy spanner constructions gives rfault tolerant kspanners with Õ(r2 1+ 2 n k+1) edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [CLPR09] depends similarly on n but exponentially on r (approximately like k r). For the case of k = 2 and unit edgelengths, an O(r log n)approximation is known from recent work of Dinitz and Krauthgamer [DK11], in which several spanner results are obtained using a common approach of rounding a natural flowbased linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to O(log n), which is, notably, independent of the number of faults r. We further strengthen this bound in terms of the maximum degree by using the Lovász Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.
Good quality virtual realization of unit ball graphs
 of Lecture Notes in Computer Science
, 2007
"... The quality of an embedding Φ: V ↦ → R 2 of a graph G = (V, E) into the Euclidean plane is the ratio of max{u,v}∈E Φ(u) − Φ(v)2 to min{u,v}�∈E Φ(u) − Φ(v)2. Given a graph G = (V, E), that is known to be a unit ball graph in fixed dimensional Euclidean space R d, we seek algorithms to compu ..."
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Cited by 13 (3 self)
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The quality of an embedding Φ: V ↦ → R 2 of a graph G = (V, E) into the Euclidean plane is the ratio of max{u,v}∈E Φ(u) − Φ(v)2 to min{u,v}�∈E Φ(u) − Φ(v)2. Given a graph G = (V, E), that is known to be a unit ball graph in fixed dimensional Euclidean space R d, we seek algorithms to compute an embedding Φ: V ↦ → R 2 of best (smallest) quality. Note that G comes with no associated geometric information and in this setting, related problems such as recognizing if G is a unit disk graph (UDG), are NPhard. While any connected unit disk graph (UDG) has a 2dimensional embedding with quality between 1/2 and 1, as far as we know, Vempala’s random projection approach (FOCS 1998) provides the best quality bound of O(log 3 n · √ log log n) for this problem. This paper presents a simple, combinatorial algorithm for computing a O(log 2.5 n)quality 2dimensional embedding of a given graph, that is known to be a UBG in fixed dimensional Euclidean space R d. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality2 embedding in R O(d log d). The first step of our algorithm constructs a “growthrestricted approximation ” of the given UBG. While such a construction is trivial if the UBG comes with a geometric representation, we are not aware of any other algorithm that can perform this step without geometric information. Construction of a growthrestricted approximation permits us to bypass the standard and costly technique of solving a linear program with exponentially many “spreading constraints. ” As a side effect of our construction, we get a constantfactor approximation to the minimum clique cover problem for UBGs, described without geometry. The second step of our algorithm combines the probabilistic decomposition of growthrestricted graphs due to Lee and Krauthgamer (STOC 2003) with Rao’s embedding algorithm for planar graphs (SoCG 1999) to obtain a (k, O ( √ log n))volume respecting embedding of growthrestricted graphs. Our problem is a version of the well known localization problem in wireless sensor networks, in which network nodes are required to compute virtual 2dimensional Euclidean coordinates given little or (as in our case) no geometric information.
Tradeoffs between Stretch Factor and Load Balancing Ratio in Routing on Growth Restricted Graphs
, 2004
"... A graph has growth rate k if the number of nodes in any subgraph with diameter r is bounded by O(r k). The communication graphs of wireless networks and peertopeer networks often have small growth rate. In this paper we study the tradeoff between two quality measures for routing in growth restrict ..."
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Cited by 11 (1 self)
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A graph has growth rate k if the number of nodes in any subgraph with diameter r is bounded by O(r k). The communication graphs of wireless networks and peertopeer networks often have small growth rate. In this paper we study the tradeoff between two quality measures for routing in growth restricted graphs. The two measures we consider are the stretch factor, which measures the lengths of the routing paths, and the load balancing ratio, which measures how evenly the traffic is distributed. We show that if the routing algorithm is required to use paths with stretch factor c, then its load balancing ratio is bounded by O((n/c) 1−1/k), where k is the graph’s growth rate. We illustrate our results by focusing on the unit disk graph for modeling wireless networks in which two nodes have direct communication if their distance is under certain threshold. We show that if the maximum density of the nodes is bounded by ρ, there exists routing scheme such that the stretch factor of routing paths is at most c, and the maximum load on the nodes is at most O(min ( � ρn/c, n/c)) times the optimum. In addition, the bound on the load balancing ratio is tight in the worst case. As a special case, when the density is bounded by a constant, the shortest path routing has a load balancing ratio of O (√n). The result extends to kdimensional unit ball graphs and graphs with growth rate k. We also discuss algorithmic issues for load balanced short path routing and for load balanced routing in spanner graphs.
Towards Fast Decentralized Construction of LocalityAware Overlay Networks
 In 26th Annual ACM SIGACTSIGOPS Symp. on Principles Of Distributed Computing (PODC
, 2007
"... We consider a large overlay network where any two nodes can communicate directly via the underlying Internet as long as the sender knows the recipient’s ipaddress. Due to the scalability requirement, the overlay network must be sparse: a given node can store at most a polylogarithmic number of ipad ..."
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Cited by 6 (2 self)
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We consider a large overlay network where any two nodes can communicate directly via the underlying Internet as long as the sender knows the recipient’s ipaddress. Due to the scalability requirement, the overlay network must be sparse: a given node can store at most a polylogarithmic number of ipaddresses. A notion of distance (locality) in the network is given by nodetonode roundtrip times. We assume that initially the overlay links are random, and hence have no explicit localityaware properties. We provide fast distributed constructions for various localityaware (lowstretch) distributed data structures, such as: distance labeling schemes, nameindependent routing schemes, and multicast trees. In previous work, such data structures have only been constructed via centralized algorithms. Our constructions complete in polylogarithmic time (and thus induce at most a polylogarithmic load on every given node), and achieve quality guarantees similar to those of the corresponding centralized algorithms. Our algorithms use a common localityaware, smallworldlike overlay framework, constructed via concurrent random walks. Our guarantees are for growthconstrained metrics, a wellstudied family of
Ultralowdimensional embeddings for doubling metrics
 IN PROCEEDINGS OF THE 19TH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2008
"... We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume” argument ..."
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Cited by 4 (0 self)
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We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume” argument shows that this bound is nearly tight: a uniform metric on n points requires nearly logarithmic number of dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to lowdimensional embeddings. In other words, do doubling metrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embeddings, embed in low dimensional Euclidean spaces with small distortion? In this paper, we give a positive answer to this question. We show how to embed any doubling metrics into O(log log n) dimensions with o(log n) distortion. This is the first embedding for doubling metrics into fewer than logarithmic number of dimensions, even allowing for logarithmic distortion. This result is one extreme point of our general tradeoff between distortion and dimension: given an npoint metric (V, d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we show that the metric embeds into Euclidean space RT with O(log n dimD /T) distortion.