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357
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
Improved approximation algorithms for minimum weight vertex separators
 In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS’89
, 1989
"... vertex separators ..."
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A Decentralized Algorithm for Spectral Analysis
, 2004
"... In many large network settings, such as computer networks, social networks, or hyperlinked text documents, much information can be obtained from the network’s spectral properties. However, traditional centralized approaches for computing eigenvectors struggle with at least two obstacles: the data ma ..."
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Cited by 94 (2 self)
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In many large network settings, such as computer networks, social networks, or hyperlinked text documents, much information can be obtained from the network’s spectral properties. However, traditional centralized approaches for computing eigenvectors struggle with at least two obstacles: the data may be difficult to obtain (both due to technical reasons and because of privacy concerns), and the sheer size of the networks makes the computation expensive. A decentralized, distributed algorithm addresses both of these obstacles: it utilizes the computational power of all nodes in the network and their ability to communicate, thus speeding up the computation with the network size. And as each node knows its incident edges, the data collection problem is avoided as well. Our main result is a simple decentralized algorithm for computing the top k eigenvectors of a symmetric weighted adjacency matrix, and a proof that it converges essentially in O(τmix log 2 n) rounds of communication and computation, where τmix is the mixing time of a random walk on the network. An additional contribution of our work is a decentralized way of actually detecting convergence, and diagnosing the current error. Our protocol scales well, in that the amount of computation performed at any node in any one round, and the sizes of messages sent, depend polynomially on k, but not at all on the (typically much larger) number n of nodes.
LowerStretch Spanning Trees
, 2005
"... ... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an nvertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds i ..."
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Cited by 86 (11 self)
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... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an nvertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds is listed as one of the longstanding open questions in the area of lowdistortion embeddings of metrics (Matousek 2002). We significantly reduce this gap by constructing a spanning tree in G of average stretch O((log n log log n) 2). Moreover, we show that this tree can be constructed in time O(m log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for diagonally dominant linear systems of Spielman and Teng from to m2 (O( √ log n log log n)) log(1/ɛ) m log O(1) n log(1/ɛ), and to O(n(log n log log n) 2 log(1/ɛ)) when the system is planar. Applying a recent reduction of Boman, Hendrickson and Vavasis, this provides an O(n(log n log log n) 2 log(1/ɛ)) time algorithm for solving the linear systems that arise when applying the finite element method to solve twodimensional elliptic partial differential equations. Our result can also be used to improve several earlier approximation algorithms that use lowstretch spanning trees.
On Certain Connectivity Properties of the Internet Topology
 IN PROC. 35TH ACM SYMP. ON THEORY OF COMPUTING
, 2003
"... We show that random graphs in the preferential connectivity model have constant conductance, and hence have worstcase routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the rando ..."
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Cited by 85 (3 self)
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We show that random graphs in the preferential connectivity model have constant conductance, and hence have worstcase routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the VickreyClarkeGroves mechanism for shortest paths) of a random graph is bounded by a small constant.
Energy Efficient Routing in Ad Hoc Disaster Recovery Networks
, 2003
"... The terrorist attacks on September 11, 2001 have drawn attention to the use of wireless technology in order to locate survivors of structural collapse. We propose to construct an ad hoc network of wireless smart badges in order to acquire information from trapped survivors. We investigate the energy ..."
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Cited by 82 (1 self)
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The terrorist attacks on September 11, 2001 have drawn attention to the use of wireless technology in order to locate survivors of structural collapse. We propose to construct an ad hoc network of wireless smart badges in order to acquire information from trapped survivors. We investigate the energy efficient routing problem that arises in such a network and show that since smart badges have very Hmited power sources and very low data rates, which may be inadequate in an emergency situation, the solution of the routing problem requires new protocols. The problem is formulated as an anycast routing problem in which the objective is to maximize the time until the first battery drainsout. We present iterative algorithms for obtaining the optimal solution of the problem. Then, we derive an upper bound on the network lifetime for specific topologies. Finally, a polynomial algorithm for obtaining the optimal solution in such topologies is described.
Conductance and Congestion in Power Law Graphs
, 2003
"... It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size ..."
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Cited by 69 (6 self)
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It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size of the network? We study routing in families of sparse random graphs whose degrees follow heavy tailed distributions. Instantiations of such random graphs have been proposed as models for the topology of the Internet at the level of Autonomous Systems as well as at the level of routers. Let n be the number of nodes. Suppose that for each pair of nodes with degrees du and dv we have O(dudv ) units of demand. Thus the total demand is O(n ). We argue analytically and experimentally that in the considered random graph model such demand patterns can be routed so that the flow through each link is at most O . This is to be compared with a bound # that holds for arbitrary graphs. Similar results were previously known for sparse random regular graphs, a.k.a. "expander graphs." The significance is that Internetlike topologies, which grow in a dynamic, decentralized fashion and appear highly inhomogeneous, can support routing with performance characteristics comparable to those of their regular counterparts, at least under the assumption of uniform demand and capacities. Our proof uses approximation algorithms for multicommodity flow and establishes strong bounds of a generalization of "expansion," namely "conductance." Besides routing, our bounds on conductance have further implications, most notably on the gap between first and second eigenvalues of the stochastic normalization of the adjacency matrix of the graph.
Balanced graph partitioning
 In 16th Annual ACM Symposium on Parallelism in Algorithms and Architectures
, 2004
"... We consider the problem of partitioning a graph into k components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter ν ≥ 1, no component contains more than ν · n k of the graph vertices. For k = 2 an ..."
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Cited by 68 (0 self)
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We consider the problem of partitioning a graph into k components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter ν ≥ 1, no component contains more than ν · n k of the graph vertices. For k = 2 and ν = 1 this problem is equivalent to the well known Minimum Bisection Problem for which an approximation algorithm with a polylogarithmic approximation guarantee has been presented in [FK02]. For arbitrary k and ν ≥ 2 a bicriteria approximation ratio of O(logn) was obtained by [ENRS99] using the spreading metrics technique. We present a bicriteria approximation algorithm that for any constant ν> 1 runs in polynomial time and guarantees an approximation ratio of O(log1.5 n) (for a precise statement of the main result see Theorem 6). For ν = 1 and k ≥ 3 we show that no polynomial time approximation algorithm can guarantee a finite approximation ratio unless P = NP. 1
Inoculation Strategies for Victims of Viruses and the SumofSquares Partition Problem (Extended Abstract)
, 2004
"... ..."
A local clustering algorithm for massive graphs and its application to nearlylinear time graph partitioning
, 2013
"... We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal conn ..."
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Cited by 60 (9 self)
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We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal connections are significantly richer than its external connections—near a given vertex. The running time of our algorithm, when it finds a nonempty local cluster, is nearly linear in the size of the cluster it outputs. The running time of our algorithm also depends polylogarithmically on the size of the graph and polynomially on the conductance of the cluster it produces. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and webgraphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly linear time algorithm for solving linear systems in symmetric, diagonally dominant matrices. The linear system solver also leads to a nearly linear time algorithm for approximating the secondsmallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.