Matrix multiplication via arithmetic progressions

Coppersmith, Don; Winograd, S.

doi:10.1145/28395.28396

Cited by 1,009 publications

(1,112 citation statements)

References 6 publications

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“…The state of the art algorithm is due to Alon, Yuster and Zwick [2] and runs in O(m 2ω ω+1 ), where currently the fast matrix multiplication exponent ω is 2.371 [10]. Thus, the Alon et al algorithm currently runs in O(m 1.41 ) time.…”

Section: Existing Workmentioning

confidence: 99%

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Kolountzakis

Peng

Tsourakakis

2010

Algorithms and Models for the Web-Graph

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Abstract. The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model [15]. The key idea of our algorithm is to combine the sampling algorithm of [34,35] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [2], treating each set appropriately. We obtain a running time O m + m 3/2 ∆ log n tǫ 2 and an ǫ approximation (multiplicative error), where n is the number of vertices, m the number of edges and ∆ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage O m 1/2 log n + m 3/2 ∆ log n tǫ 2 and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.

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Section: Existing Workmentioning

confidence: 99%

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Kolountzakis

Peng

Tsourakakis

2010

Algorithms and Models for the Web-Graph

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“…The generic algorithm has a number of instantiations, depending on how its steps are implemented (in the RAM or I/O model). Below, we compare the worst-case performance analysis of Algorithm 1 in the RAM model with the analysis of the classical sort-merge-join, the results by Coppersmith and Winograd [3] and Yuster and Zwick [9]. We emphasize analysis because the various analyses are not tight to the actual performance of the algorithms.…”

Section: Related Workmentioning

confidence: 99%

“…As explained in more detail later, one way to improve the worst-case behavior in cases similar to the example above is to represent the input tuples of R1 and R2 as adjacency matrices of size n×n and construct the result by multiplying the matrices inÕ(n 2.376 ) time [3].…”

Section: Introductionmentioning

confidence: 99%

Faster join-projects and sparse matrix multiplications

Amossen

Pagh

2009

Proceedings of the 12th International Conference on Database Theory

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Computing an equi-join followed by a duplicate eliminating projection is conventionally done by performing the two operations in serial. If some join attribute is projected away the intermediate result may be much larger than both the input and the output, and the computation could therefore potentially be performed faster by a direct procedure that does not produce such a large intermediate result. We present a new algorithm that has smaller intermediate results on worst-case inputs, and in particular is more efficient in both the RAM and I/O model. It is easy to see that join-project where the join attributes are projected away is equivalent to boolean matrix multiplication. Our results can therefore also be interpreted as improved sparse, output-sensitive matrix multiplication.

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“…For k-vertex connectivity of a graph q, Eppstein et al [10] have shown that g k ( k jI f j ) is a strong, sparse certificate of q, where f j is a breadth-first search forest in graph q À jÀI iI f i . However, currently, the number of processors used in any NC algorithm for finding a breadth-first search forest is no better than the number of scalar operations for matrix multiplication, which is yn PXQUT [7]. To reduce the number of processors used, we adopt an alternative approach to find a sparse k-vertex certificate and show the found certificate is a strong certificate.…”

Section: Maintaining 3-edge Connected Componentsmentioning

confidence: 99%

Fully dynamic maintenance of k-connectivity in parallel

Liang

Brent

Shen

2001

IEEE Trans. Parallel Distrib. Syst.

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AbstractÐGiven a graph q Y i with n vertices and m edges, the k-connectivity of q denotes either the k-edge connectivity or the k-vertex connectivity of q. In this paper, we deal with the fully dynamic maintenance of k-connectivity of q in the parallel setting for k PY Q. We study the problem of maintaining k-edge/vertex connected components of a graph undergoing repeatedly dynamic updates, such as edge insertions and deletions, and answering the query of whether two vertices are included in the same k-edge/ vertex connected component. Our major results are the following: 1) An NC algorithm for the 2-edge connectivity problem is proposed, which runs in ylog n logman time using yn QaR processors per update and query.2) It is shown that the biconnectivity problem can be solved in ylog P n time using ynPnY na log n processors per update and yI time with a single processor per query or in ylog n log m n time using ynPnY na log n processors per update and ylog n time using ynPnY na log n processors per query, where XY X is the inverse of Ackermann's function. 3) An NC algorithm for the triconnectivity problem is also derived, which takes ylog n log m n log n log log naQnY n time using ynQnY na log n processors per update and yI time with a single processor per query. 4) An NC algorithm for the 3-edge connectivity problem is obtained, which has the same time and processor complexities as the algorithm for the triconnectivity problem. To the best of our knowledge, the proposed algorithms are the first NC algorithms for the problems using yn processors in contrast to m processors for solving them from scratch. In particular, the proposed NC algorithm for the 2-edge connectivity problem uses only yn QaR processors. All the proposed algorithms run on a CRCW PRAM.Index TermsÐNC algorithms, 2-edge/vertex connectivity, 3-edge/vertex connectivity, dynamic data structures, parallel algorithm design and analysis, graph problems.

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Matrix multiplication via arithmetic progressions

Cited by 1,009 publications

References 6 publications

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

Faster join-projects and sparse matrix multiplications

Fully dynamic maintenance of k-connectivity in parallel

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