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34
Graph sketches: sparsification, spanners, and subgraphs
 In PODS
, 2012
"... When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., tho ..."
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Cited by 46 (10 self)
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When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., those based on linear projections of the data. These are applicable in many models including various parallel, stream, and compressed sensing settings. A rich body of analytic and empirical work exists for sketching numerical data such as the frequencies of a set of entities. Our work investigates graph sketching where the graphs of interest encode the relationships between these entities. The main challenge is to capture this richer structure and build the necessary synopses with only linear measurements. In this paper we consider properties of graphs including the size of the cuts, the distances between nodes, and the prevalence of
Low Distortion Spanners
"... A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an fspanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tr ..."
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Cited by 26 (3 self)
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A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an fspanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood. In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2 and 6spanners of Aingworth et al. and Baswana et al. and improve on the (1+ɛ, β)spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially.
On the locality of distributed sparse spanner construction
 In ACM Press, editor, 27th Annual ACM Symp. on Principles of Distributed Computing (PODC
, 2008
"... The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)spanner of O(kn 1+1/k)edgesforeverynnode unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)spanner with O ..."
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Cited by 17 (6 self)
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The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)spanner of O(kn 1+1/k)edgesforeverynnode unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)spanner with O(kn 1+1/k) edges.) Previous distributed solutions achieving such optimal stretchsize tradeoff either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ɛ>0, constructs a (1 + ɛ, 2)spanner of O(ɛ −1 n 3/2)edgesin O(ɛ −1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k − 1, 0)spanner of o(n 1+1/(k−1))edgesfork ∈{2, 3, 5}. It is also shown that for every k>1, any (randomized) distributed algorithm that constructs a spanner with fewer than n 1+1/k+ɛ edges in at most n ɛ expected rounds must stretch some distances by an additive factor of n Ω(ɛ).Inotherwords, while additive stretched spanners with O(n 1+1/k) edges may exist, e.g., for k =2, 3, they cannot be computed distributively in a subpolynomial number of rounds in expectation. Supported by the équipeprojet INRIA “DOLPHIN”. Supported by the ANRproject “ALADDIN”, and the
Additive Spanners and (α, β)Spanners
"... An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). How ..."
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Cited by 14 (3 self)
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An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)spanners. Our first result is an additive (1, 6)spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively wellapproximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparsenessdistortion tradeoffs. Our second result addresses the problem of which (α, β)spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
Approximate distance oracles with improved query time
 MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT, OPEN UNIVERSITY OF ISRAEL
, 2011
"... Given an undirected graph G with m edges, n vertices, and nonnegative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)approximate distance oracle for G of size O(kn 1+1/k) can be constructed in O ( √ km+kn 1+c/ √ k) time and can answer queries in O(k ..."
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Cited by 12 (1 self)
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Given an undirected graph G with m edges, n vertices, and nonnegative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)approximate distance oracle for G of size O(kn 1+1/k) can be constructed in O ( √ km+kn 1+c/ √ k) time and can answer queries in O(k) time. We also give an oracle which is faster for smaller k. Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all k ≥ 6 and improve the O(kmn 1/k) time bound of Thorup and Zwick except for very sparse graphs and small k. When m = Ω(n 1+c/ √ k) and k = O(1), our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erdős.
Streaming algorithm for graph spanners  single pass and constant processing time per edge
 Inf. Process. Lett
"... A spanner is a sparse subgraph of a given graph that preserves approximate distance between each pair of vertices. In precise words, a tspanner of a graph G = (V,E), for any t ∈ N, is a subgraph (V,ES), ES ⊆ E ..."
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Cited by 11 (1 self)
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A spanner is a sparse subgraph of a given graph that preserves approximate distance between each pair of vertices. In precise words, a tspanner of a graph G = (V,E), for any t ∈ N, is a subgraph (V,ES), ES ⊆ E
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
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Cited by 10 (0 self)
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We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to loglogarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.
Fast Routing Table Construction Using Small Messages. ArXiv eprints
, 2012
"... We describe a distributed randomized algorithm to construct routing tables. Given 0 < ε ≤ 1/2, the algorithm runs in time Õ(n1/2+ε + HD), where n is the number of nodes and HD denotes the diameter of the network in hops (i.e., as if the network is unweighted). The weighted length of the produce ..."
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Cited by 9 (5 self)
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We describe a distributed randomized algorithm to construct routing tables. Given 0 < ε ≤ 1/2, the algorithm runs in time Õ(n1/2+ε + HD), where n is the number of nodes and HD denotes the diameter of the network in hops (i.e., as if the network is unweighted). The weighted length of the produced routes is at most O(ε−1 log ε−1) times the optimal weighted length. This is the first algorithm to break the Ω(n) complexity barrier for computing weighted shortest paths even for a single source. Moreover, the algorithm nearly meets the Ω̃(n1/2 + HD) lower bound for distributed computation of routing tables and approximate distances (with optimality, up to polylog factors, for ε = 1 / logn). The presented techniques have many applications, including improved distributed approximation algorithms for Generalized Steiner Forest, allpairs distance estimation, and estimation of the weighted diameter.
FASTER ALGORITHMS FOR ALLPAIRS APPROXIMATE SHORTEST PATHS IN UNDIRECTED GRAPHS
, 2006
"... Let G = (V, E) be a weighted undirected graph having nonnegative edge weights. An estimate ˆ δ(u, v) of the actual distance δ(u, v) between u, v ∈ V is said to be of stretch t iff δ(u, v) ≤ ˆ δ(u, v) ≤ t · δ(u, v). Computing allpairs small stretch distances efficiently (both in terms of time ..."
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Cited by 9 (2 self)
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Let G = (V, E) be a weighted undirected graph having nonnegative edge weights. An estimate ˆ δ(u, v) of the actual distance δ(u, v) between u, v ∈ V is said to be of stretch t iff δ(u, v) ≤ ˆ δ(u, v) ≤ t · δ(u, v). Computing allpairs small stretch distances efficiently (both in terms of time and space) is a wellstudied problem in graph algorithms. We present a simple, novel and generic scheme for allpairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for allpairs tstretch distances for a whole range of stretch t, and also answer an open question posed by Thorup and Zwick in their
Distributed approximation algorithms for weighted shortest paths
 In Proc. of the Symp. on Theory of Comp. (STOC
"... A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O(log n) bandwidth restriction on edges (the standard synchronous CONGEST model). In this problem ..."
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A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O(log n) bandwidth restriction on edges (the standard synchronous CONGEST model). In this problem every node wants to know its distance to some others nodes. The question whether approximation algorithms help speed up the shortest paths computation (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is wellunderstood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both singlesource shortest paths (SSSP) and allpairs shortest paths (APSP) in the weighted case. Our main result is an algorithm for SSSP. Previous results are the classic O(n)time BellmanFord algorithm and an Õ(n1/2+1/2k + D)time (8kdlog(k + 1)e − 1)approximation algorithm, for any integer k ≥ 1, which follows from the result of Lenzen and PattShamir (STOC 2013). (Note that Lenzen and PattShamir in fact solve a harder problem, and we use Õ(·) to hide the