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17
Graph distances in the streaming model: the value of space
 In ACMSIAM Symposium on Discrete Algorithms
, 2005
"... We investigate the importance of space when solving problems based on graph distance in the streaming model. In this model, the input graph is presented as a stream of edges in an arbitrary order. The main computational restriction of the model is that we have limited space and therefore cannot stor ..."
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Cited by 52 (10 self)
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We investigate the importance of space when solving problems based on graph distance in the streaming model. In this model, the input graph is presented as a stream of edges in an arbitrary order. The main computational restriction of the model is that we have limited space and therefore cannot store all the streamed data; we are forced to make spaceefficient summaries of the data as we go along. For a graph of n vertices and m edges, we show that testing many graph properties, including connectivity (ergo any reasonable decision problem about distances) and bipartiteness, requires Ω(n) bits of space. Given this, we then investigate how the power of the model increases as we relax our space restriction. Our main result is an efficient randomized algorithm that constructs a (2t + 1)spanner in one pass. With high probability, it uses O(t · n 1+1/t log 2 n) bits of space and processes each edge in the stream in O(t 2 · n 1/t log n) time. We find approximations to diameter and girth via the log n constructed spanner. For t = Ω (), the space log log n requirement of the algorithm is O(n·polylog n), and the peredge processing time is O(polylog n). We also show a corresponding lower bound of t for the approximation ratio achievable when the space restriction is O(t · n1+1/t log 2 n). We then consider the scenario in which we are allowed multiple passes over the input stream. Here, we investigate whether allowing these extra passes will compensate for a given space restriction. We show that ∗This work was supported by the DoD University Research Initiative (URI) administered by the Office of Naval Research
New Constructions of (α, β)Spanners and Purely Additive Spanners
, 2005
"... An ¦ α § β ¨spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨spanners of size ..."
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Cited by 24 (6 self)
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An ¦ α § β ¨spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨spanners of size O ¦ n 1 � 1 � k ¨ and that there exist (purely additive) ¦ 1 § 2 ¨spanners of size O ¦ n 3 � 2 ¨. However no other ¦ 1 § O ¦ 1¨� ¨spanners are known to exist. In this paper we develop a couple new techniques for constructing ¦ α § β ¨spanners. The first result is a purely additive ¦ 1 § 6 ¨spanner of size O ¦ n 4 � 3 ¨. Our 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. This general approach should lead to new spanner constructions. The second result is a truly simple linear time construction of ¦ k § k � 1 ¨spanners with size O ¦ n 1 � 1 � k ¨. In a distributed network the algorithm terminates in a constant number of rounds and has expected size O ¦ n 1 � 1 � k ¨. The new idea here is primarily in the analysis of the construction. We show that a few simple and local rules for picking spanner edges induce seemingly coordinated global behavior.
GRAPH DISTANCES IN THE DATASTREAM MODEL
, 2008
"... We explore problems related to computing graph distances in the datastream model. The goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory. We are motivated by both the practical challenge of processing massive graph ..."
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Cited by 20 (3 self)
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We explore problems related to computing graph distances in the datastream model. The goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory. We are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the datastream model. In particular, we are interested in the tradeoffs between model parameters such as perdataitem processing time, total space, and the number of passes that may be taken over the stream. These tradeoffs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature. Our results include the following: (1) Spanner construction: There exists a singlepass, Õ(tn1+1/t)space, Õ(t2n1/t)timeperedge algorithm that constructs a (2t + 1)spanner. For t =Ω(logn/log log n), the algorithm satisfies the semistreaming space restriction of O(n polylog n) and has peredge processing time O(polylog n). This resolves an open question from [J. Feigenbaum et al., Theoret. Comput. Sci., 348 (2005), pp. 207–216]. (2) Breadthfirstsearch (BFS) trees: For any even constant k, we show that any algorithm that computes the first k layers of a BFS tree from a prescribed node with probability at least 2/3 requires either greater than k/2 passes or ˜Ω(n1+1/k) space. Since constructing BFS trees is
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 16 (7 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
Low Distortion Spanners
"... Abstract. 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 clearl ..."
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Cited by 16 (2 self)
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Abstract. 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. 1
Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners
 In International Colloquium on Automata, Languages and Programming
, 2007
"... Abstract. We present a streaming algorithm for constructing sparse spanners and show that our algorithm outperforms significantly the stateoftheart algorithm for this task [20]. Specifically, the processing timeperedge of our algorithm is drastically smaller than that of the algorithm of [20], ..."
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Cited by 10 (2 self)
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Abstract. We present a streaming algorithm for constructing sparse spanners and show that our algorithm outperforms significantly the stateoftheart algorithm for this task [20]. Specifically, the processing timeperedge of our algorithm is drastically smaller than that of the algorithm of [20], and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters for the stateoftheart algorithm. We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has a very small incremental update time, and a nontrivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides nontrivial bounds on both incremental and decremental update time for a wide range of stretch parameter t. 1
Fast deterministic distributed algorithms for sparse spanners
 IN 13 TH INTERNATIONAL COLLOQUIUM ON STRUCTURAL INFORMATION & COMMUNICATION COMPLEXITY (SIROCCO
, 2006
"... This paper concerns the efficient construction of sparse and low stretch spanners for unweighted arbitrary graphs with n nodes. All previous deterministic distributed algorithms, for constant stretch spanner of o(n²) edges, have a running time Ω(n^ɛ) for some constant ɛ > 0 depending on the stretch. ..."
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Cited by 7 (3 self)
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This paper concerns the efficient construction of sparse and low stretch spanners for unweighted arbitrary graphs with n nodes. All previous deterministic distributed algorithms, for constant stretch spanner of o(n²) edges, have a running time Ω(n^ɛ) for some constant ɛ > 0 depending on the stretch. Our deterministic distributed algorithms construct constant stretch spanners of o(n²) edges in o(n^ɛ) time for any constant ɛ > 0. More precisely, in the Linial’s free model, we construct in n O(1/ √ log n) time, for every graph, a 5spanner of O(n 3/2) edges. The result is extended to O(k 2.322)spanners with O(n 1+1/k) edges for every parameter k � 1. If the minimum degree of the graph is Ω(√n), then, in the same time complexity, a 9spanner with O(n) edges can be constructed.
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 7 (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. 1
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 6 (2 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.
Local Computation of Nearly Additive Spanners
"... Abstract. An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a spar ..."
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Cited by 6 (3 self)
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Abstract. An (α, β)spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every nnode graph and integer k ≥ 1, an (α, β)spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic