Further Algebraic Algorithms in the Congested Clique Model and Applications to Graph-Theoretic Problems

Censor-Hillel et al. [PODC’15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorithms, in the congested clique model, for efficiently computing multiple independent instances of matrix products, computing the determinant, the rank and the inverse of a matrix, and solving systems of linear equations. As applications of these techniques, we obtain more efficient algorithms for the computation, again in the congested clique model, of the all-pairs shortest paths and the diameter in directed and undirected graphs with small weights, improving over Censor-Hillel et al.’s work. We also obtain algorithms for several other graph-theoretic problems such as computing the number of edges in a maximum matching and the Gallai-Edmonds decomposition of a simple graph, and computing a minimum vertex cover of a bipartite graph.

[1]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[2]  J. Hopcroft,et al.  Triangular Factorization and Inversion by Fast Matrix Multiplication , 1974 .

[3]  L. Csanky,et al.  Fast parallel matrix inversion algorithms , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[4]  Franco P. Preparata,et al.  An Improved Parallel Processor Bound in Fast Matrix Inversion , 1978, Inf. Process. Lett..

[5]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[6]  László Lovász,et al.  On determinants, matchings, and random algorithms , 1979, FCT.

[7]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[8]  Douglas H. Wiedemann.  Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[9]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[10]  Vijay V. Vazirani,et al.  Maximum Matchings in General Graphs Through Randomization , 1989, J. Algorithms.

[11]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[12]  Erich Kaltofen,et al.  On Wiedemann's Method of Solving Sparse Linear Systems , 1991, AAECC.

[13]  Wang Wei.  On the least prime in an arithmetic progression , 1991 .

[14]  Victor Y. Pan,et al.  Processor efficient parallel solution of linear systems over an abstract field , 1991, SPAA '91.

[15]  Victor Y. Pan,et al.  Processor-efficient parallel solution of linear systems. II. The positive characteristic and singular cases , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[16]  Raimund Seidel,et al.  On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs , 1995, J. Comput. Syst. Sci..

[17]  J. Cheriyan.  Randomized $\tilde{O}(M( , 1997 .

[18]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[19]  Noga Alon,et al.  On the Exponent of the All Pairs Shortest Path Problem , 1991, J. Comput. Syst. Sci..

[20]  J. Cheriyan.  Randomized Õ(M(|V|)) Algorithms for Problems in Matching Theory , 1997, SIAM J. Comput..

[21]  Tadao Takaoka,et al.  Subcubic Cost Algorithms for the All Pairs Shortest Path Problem , 1998, Algorithmica.

[22]  Uri Zwick,et al.  All pairs shortest paths in undirected graphs with integer weights , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[23]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[24]  B. D. Saunders,et al.  Efficient matrix preconditioners for black box linear algebra , 2002 .

[25]  Uri Zwick,et al.  All pairs shortest paths using bridging sets and rectangular matrix multiplication , 2000, JACM.

[26]  Boaz Patt-Shamir,et al.  MST construction in O(log log n) communication rounds , 2003, SPAA '03.

[27]  Piotr Sankowski,et al.  Maximum matchings via Gaussian elimination , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[28]  Georg Schnitger,et al.  Communication complexity of matrix computation over finite fields , 1995, Mathematical systems theory.

[29]  Anindya De,et al.  Fast integer multiplication using modular arithmetic , 2008, STOC.

[30]  Roger Wattenhofer,et al.  Tight bounds for parallel randomized load balancing: extended abstract , 2011, STOC '11.

[31]  Boaz Patt-Shamir,et al.  The round complexity of distributed sorting: extended abstract , 2011, PODC '11.

[32]  Jane Zundel.  MATCHING THEORY , 2011 .

[33]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[34]  Piotr Sankowski,et al.  Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[35]  François Le Gall,et al.  Faster Algorithms for Rectangular Matrix Multiplication , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[36]  Christoph Lenzen,et al.  "Tri, Tri Again": Finding Triangles and Small Subgraphs in a Distributed Setting - (Extended Abstract) , 2012, DISC.

[37]  Anindya De,et al.  Fast Integer Multiplication Using Modular Arithmetic , 2013, SIAM J. Comput..

[38]  Christoph Lenzen,et al.  Optimal deterministic routing and sorting on the congested clique , 2012, PODC '13.

[39]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[40]  Fabian Kuhn,et al.  On the power of the congested clique model , 2014, PODC.

[41]  Ryan Williams,et al.  Faster all-pairs shortest paths via circuit complexity , 2013, STOC.

[42]  Danupon Nanongkai,et al.  Distributed approximation algorithms for weighted shortest paths , 2014, STOC.

[43]  Sriram V. Pemmaraju,et al.  Near-Constant-Time Distributed Algorithms on a Congested Clique , 2014, DISC.

[44]  Sriram V. Pemmaraju,et al.  Lessons from the Congested Clique applied to MapReduce , 2015, Theor. Comput. Sci..

[45]  Sriram V. Pemmaraju,et al.  Toward Optimal Bounds in the Congested Clique: Graph Connectivity and MST , 2015, PODC.

[46]  Monika Henzinger,et al.  A deterministic almost-tight distributed algorithm for approximating single-source shortest paths , 2015, STOC.

[47]  Christoph Lenzen,et al.  Algebraic methods in the congested clique , 2015, Distributed Computing.