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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
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 28 (7 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
Faulttolerant spanners for general graphs
 in STOC’09, 2009
"... The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected nvertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a kspanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u ..."
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Cited by 17 (4 self)
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The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected nvertex graph G = (V,E) and an integer k ≥ 1, the subgraph H = (V,E′), E ′ ⊆ E, is a spanner of stretch k (or, a kspanner) of G if δH(u, v) ≤ k · δG(u, v) for every u, v ∈ V, where δG′(u, v) denotes the distance between u and v in G Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k−1)spanner of size O(n1+1/k), and this sizestretch tradeoff is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [Levcopoulos et al., STOC’98]. A subgraph H is an fvertex fault tolerant kspanner of the graph G if for any set F ⊆ V of size at most f and any pair of vertices u, v ∈ V \ F, the distances in H satisfy δH\F (u, v) ≤ k · δG\F (u, v). Levcopoulos et al. presented an efficient algorithm that given a set S of n points in Rd, constructs an fvertex fault tolerant geometric (1+)spanner for S, that is, a sparse graph H such that for every set F ⊆ S of size f and any pair of points u, v ∈ S \F, δH\F (u, v) ≤ (1+)uv, where uv  is the Euclidean distance between u and v. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [Czumaj & Zhao, SoCG’03]. This paper also raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph. The current paper answers this question in the affirmative, presenting an fvertex fault tolerant (2k−1)spanner of size
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.
A survey on streaming algorithms for massive graphs
 Managing and Mining Graph Data, volume 40 of Advances in Database Systems
, 2010
"... ..."
A nearoptimal distributed fully dynamic algorithm for maintaining sparse spanners
 Proceedings of the twentysixth annual ACM symposium on Principles of distributed computing
, 2006
"... Currently, there are no known explicit algorithms for the great majority of graph problems in the dynamic distributed messagepassing model. Instead, most stateoftheart dynamic distributed algorithms are constructed by composing a static algorithm for the problem at hand with a simulation techniq ..."
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Cited by 7 (1 self)
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Currently, there are no known explicit algorithms for the great majority of graph problems in the dynamic distributed messagepassing model. Instead, most stateoftheart dynamic distributed algorithms are constructed by composing a static algorithm for the problem at hand with a simulation technique that converts static algorithms to dynamic ones. We argue that this powerful methodology does not provide satisfactory solutions for many important dynamic distributed problems, and this necessitates developing algorithms for these problems from scratch. In this paper we develop a fully dynamic distributed algorithm for maintaining sparse spanners. Our algorithm improves drastically the quiescence time of the stateoftheart algorithm for the problem. Moreover, we show that the quiescence time of our algorithm is optimal up to a small constant factor. In addition, our algorithm improves significantly upon the stateoftheart algorithm in all efficiency parameters, specifically, it has smaller quiescence message and space complexities, and smaller local processing time. Finally, our algorithm is selfcontained and fairly simple, and is, consequently, amenable to implementation on unsophisticated network devices.
Single Pass Spectral Sparsification in Dynamic Streams
"... We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semistreaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph G, our algorithm maintains a randomized linear sketch of the incidence matrix of G in ..."
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Cited by 7 (1 self)
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We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semistreaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph G, our algorithm maintains a randomized linear sketch of the incidence matrix of G into dimension O ( 1 ɛ 2 n polylog(n)). Using this sketch, at any point, the algorithm can output a (1 ± ɛ) spectral sparsifier for G with high probability. While O ( 1 ɛ 2 n polylog(n)) space algorithms are known for computing cut sparsifiers in dynamic streams [AGM12b, GKP12] and spectral sparsifiers in insertiononly streams [KL11], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension Ω ( 1 ɛ 2 n 5/3) [AGM14]. To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of G’s incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral
Graph stream algorithms: A survey
, 2013
"... Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was twofold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine ..."
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Cited by 6 (1 self)
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Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was twofold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the stateoftheart results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas. 1.