Streaming algorithm for graph spanners—single pass and constant processing time per edge

Baswana, Surender

doi:10.1016/j.ipl.2007.11.001

Cited by 55 publications

(43 citation statements)

References 12 publications

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“…Streaming algorithms for constructing multiplicative spanners were given in [38,31,14], and nearadditive spanners in [29,37]. Spanners and emulators with sublinear error were given in [57,51].…”

Section: Related Workmentioning

confidence: 99%

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Elkin

Neiman

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

155

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Miller et al. [43] devised a distributed 1 algorithm in the CONGEST model, that given a parameter k = 1, 2, . . ., constructs an O(k)-spanner of an input unweighted nvertex graph with O(n 1+1/k ) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k − 1)-spanner with O(n 1+1/k / ) edges, with probability 1 − , for any > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 − o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm.We also devise improved bounds (and algorithms realizing these bounds) for (1 + , β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any > 0, there exists a (1 + , ( log log n ) log log n )-emulator with O(n) edges. All previous constructions of (1 + , β)-spanners and emulators employ a superlinear number of edges, for all choices of parameters.Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs. * Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. 1 They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in [43].

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Section: Related Workmentioning

confidence: 99%

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Elkin

Neiman

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

155

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“…Since the preliminary version of this paper appeared, improved spanner construction algorithms were presented by Baswana [8] and Elkin [18]. Extensions of ideas in section 6 have been developed by Zelke [41,42].…”

Section: Techniques For Decreasing Per-edge Processingmentioning

confidence: 99%

Graph Distances in the Data-Stream Model

Feigenbaum¹,

Kannan²,

McGregor³

et al. 2009

SIAM J. Comput.

137

180

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Abstract. We explore problems related to computing graph distances in the data-stream 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 trade-offs between model parameters such as perdata-item processing time, total space, and the number of passes that may be taken over the stream. These trade-offs 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 single-pass,Õ(tn 1+1/t )-space,Õ(t 2 n 1/t )-time-per-edge algorithm that constructs a (2t + 1)-spanner. For t = Ω(log n/log log n), the algorithm satisfies the semistreaming space restriction of O(n polylog n) and has per-edge processing time O(polylog n).This resolves an open question from [J. Feigenbaum et al., Theoret. Comput. Sci., 348 (2005), pp. 207-216]. (2) Breadth-first-search (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 orΩ(n 1+1/k ) space. Since constructing BFS trees is an important subroutine in many traditional graph algorithms, this demonstrates the need for new algorithmic techniques when processing graphs in the data-stream model. (3) Graph-distance lower bounds: Any t-approximation of the distance between two nodes requires Ω(n 1+1/t ) space. We also prove lower bounds for determining the length of the shortest cycle and other graph properties. (4) Techniques for decreasing per-edge processing: We discuss two general techniques for speeding up the per-edge computation time of streaming algorithms while increasing the space by only a small factor.

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“…For the problem of approximating the maximum weighted matching in a graph, a series of papers [13,26,29,12] reduced the best known approximation ratio for a semi-streaming algorithm to 4.91 + ε. There are randomized streaming algorithms that estimate the number of triangles in a graph [5,22,10] and that compute a graph spanner to approximate pairwise vertex distances [11,6].…”

Section: Introductionmentioning

confidence: 99%