Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Distributed Computing (DISC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9888))

Included in the following conference series:

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer, Heidelberg (1997)

    Book  MATH  Google Scholar 

  2. Censor-Hillel, K., Kaski, P., Korhonen, J.H., Lenzen, C., Paz, A., Suomela, J.: Algebraic methods in the congested clique. In: Proceedings of the 34th Symposium on Principles of Distributed Computing, pp. 143–152 (2015)

    Google Scholar 

  3. Cheriyan, J.: Randomized \(\tilde{O}(M(|V|))\) algorithms for problems in matching theory. SIAM J. Comput. 26(6), 1635–1669 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chu, J.I., Schnitger, G.: Communication complexity of matrix computation over finite fields. Math. Syst. Theor. 28(3), 215–228 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Csanky, L.: Fast parallel matrix inversion algorithms. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pp. 11–12 (1975)

    Google Scholar 

  6. Dolev, D., Lenzen, C., Peled, S.: “Tri, tri again”: finding triangles and small subgraphs in a distributed setting. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 195–209. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  7. Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, pp. 367–376 (2014)

    Google Scholar 

  8. Hegeman, J.W., Pandurangan, G., Pemmaraju, S.V., Sardeshmukh, V.B., Scquizzato, M.: Toward optimal bounds in the congested clique: Graph connectivity and MST. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, pp. 91–100 (2015)

    Google Scholar 

  9. Hegeman, J.W., Pemmaraju, S.V.: Lessons from the congested clique applied to mapreduce. In: Halldórsson, M.M. (ed.) SIROCCO 2014. LNCS, vol. 8576, pp. 149–164. Springer, Heidelberg (2014)

    Google Scholar 

  10. Hegeman, J.W., Pemmaraju, S.V., Sardeshmukh, V.B.: Near-constant-time distributed algorithms on a congested clique. In: Kuhn, F. (ed.) DISC 2014. LNCS, vol. 8784, pp. 514–530. Springer, Heidelberg (2014)

    Google Scholar 

  11. Henzinger, M., Krinninger, S., Nanongkai, D.: A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. In: Proceedings of the 48th Annual ACM Symposium on Theory of Computing, pp. 489–498 (2016)

    Google Scholar 

  12. Kaltofen, E., Pan, V.Y.: Processor efficient parallel solution of linear systems over an abstract field. In: Proceedings of the 3rd Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 180–191 (1991)

    Google Scholar 

  13. Kaltofen, E., Pan, V.Y.: Processor-efficient parallel solution of linear systems II: the positive characteristic and singular cases (extended abstract). In: Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pp. 714–723 (1992)

    Google Scholar 

  14. Kaltofen, E., Saunders, B.D.: On Wiedemann’s method of solving sparse linear systems. In: Proceedings of the 9th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 29–38 (1991)

    Google Scholar 

  15. Le Gall, F.: Faster algorithms for rectangular matrix multiplication. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pp. 514–523 (2012)

    Google Scholar 

  16. Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pp. 296–303 (2014)

    Google Scholar 

  17. Lenzen, C.: Optimal deterministic routing and sorting on the congested clique. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, pp. 42–50 (2013)

    Google Scholar 

  18. Lenzen, C., Wattenhofer, R.: Tight bounds for parallel randomized load balancing: extended abstract. In: Proceedings of the 43rd ACM Symposium on Theory of Computing, pp. 11–20 (2011)

    Google Scholar 

  19. Lotker, Z., Pavlov, E., Patt-Shamir, B., Peleg, D.: MST construction in \(o(\log \log n)\) communication rounds. In: Proceedings of the Fifteenth Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 94–100 (2003)

    Google Scholar 

  20. Lovász, L.: On determinants, matchings, and random algorithms. In: Fundamentals of Computation Theory, pp. 565–574 (1979)

    Google Scholar 

  21. Lovász, L., Plummer, M.D.: Matching Theory. American Mathematical Society (2009)

    Google Scholar 

  22. Nanongkai, D.: Distributed approximation algorithms for weighted shortest paths. In: Proceedings of the 46th Symposium on Theory of Computing, pp. 565–573 (2014)

    Google Scholar 

  23. Patt-Shamir, B., Teplitsky, M.: The round complexity of distributed sorting: extended abstract. In: Proceedings of the 30th Annual ACM Symposium on Principles of Distributed Computing, pp. 249–256 (2011)

    Google Scholar 

  24. Peleg, D.: Distributed computing: a locality-sensitive approach. Society for Industrial and Applied Mathematics (2000)

    Google Scholar 

  25. Preparata, F.P., Sarwate, D.V.: An improved parallel processor bound in fast matrix inversion. Inf. Process. Lett. 7(3), 148–150 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rabin, M.O., Vazirani, V.V.: Maximum matchings in general graphs through randomization. J. Algorithms 10(4), 557–567 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51(3), 400–403 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pp. 605–615 (1999)

    Google Scholar 

  29. Vassilevska Williams, V.: Multiplying matrices faster than coppersmith-winograd. In: Proceedings of the 44th Symposium on Theory of Computing, pp. 887–898 (2012)

    Google Scholar 

  30. Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49(3), 289–317 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to Arne Storjohann for precious help concerning the computation of the determinant and to anonymous reviewers for their comments. This work is supported by the Grant-in-Aid for Young Scientists (A) No. 16H05853, the Grant-in-Aid for Scientific Research (A) No. 16H01705, and the Grant-in-Aid for Scientific Research on Innovative Areas No. 24106009 of the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology in Japan.

Author information

Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Kyoto, Japan

    François Le Gall

Authors
  1. François Le Gall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Le Gall .

Editor information

Editors and Affiliations

  1. LABRI, Univ of Bordeaux LABRI, Talence Cedex, France

    Cyril Gavoille

  2. CNRS, Universite Bordeaux 1 CNRS, Talence Cedex, France

    David Ilcinkas

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Le Gall, F. (2016). Further Algebraic Algorithms in the Congested Clique Model and Applications to Graph-Theoretic Problems. In: Gavoille, C., Ilcinkas, D. (eds) Distributed Computing. DISC 2016. Lecture Notes in Computer Science(), vol 9888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53426-7_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53426-7_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53425-0

  • Online ISBN: 978-3-662-53426-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation