Skip to main content
SpringerLink
Log in

On the Communication Complexity of Linear Algebraic Problems in the Message Passing Model

  • Conference paper
Distributed Computing (DISC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8784))

Included in the following conference series:

Abstract

We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized s-player communication complexity of these problems is at least s times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, we can show the hardest input distribution is a symmetric distribution, and therefore apply a recent multi-player lower bound technique of Phillips et al. Further, we give new two-player lower bounds for a number of these problems. In particular, our optimal lower bound for the two-player version of the matrix rank problem resolves an open question of Sun and Wang.

A common feature of our lower bounds is that they apply even to the special “threshold promise” versions of these problems, wherein the underlying quantity, e.g., rank, is promised to be one of just two values, one on each side of some critical threshold. These kinds of promise problems are commonplace in the literature on data streaming as sources of hardness for reductions giving space lower bounds.

Full version available at arXiv:1407.4755 [cs.CC].

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon, N., Matias, Y., Szegedy, M.: The Space Complexity of Approximating the Frequency Moments. J. Comput. Syst. Sci. 58(1), 137–147 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: 27th Annual Symposium on Foundations of Computer Science, pp. 337–347 (1986)

    Google Scholar 

  3. Braverman, M., Ellen, F., Oshman, R., Pitassi, T., Vaikuntanathan, V.: A tight bound for set disjointness in the message-passing model. In: FOCS, pp. 668–677 (2013)

    Google Scholar 

  4. Chu, J.I., Schnitger, G.: Communication complexity of matrix computation over finite fields. Theory of Computing Systems 28, 215–228 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Clarkson, K.L., Woodruff, D.P.: Numerical linear algebra in the streaming model. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 205–214. ACM, New York (2009)

    Google Scholar 

  6. Cormode, G., Muthukrishnan, S., Yi, K.: Algorithms for distributed functional monitoring. In: Proc. 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1076–1085 (2008)

    Google Scholar 

  7. Frieze, A., Kannan, R., Vempala, S.: Fast Monte-Carlo algorithms for finding low-rank approximations. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, pp. 370–378. IEEE (1998)

    Google Scholar 

  8. Henzinger, M.R., Raghavan, P., Rajagopalan, S.: Computing on data streams. In: External Memory Algorithms: Dimacs Workshop External Memory and Visualization, May 20-22, vol. 50, p. 107. American Mathematical Soc. (1999)

    Google Scholar 

  9. Huang, W., Shi, Y., Zhang, S., Zhu, Y.: The communication complexity of the hamming distance problem. Inf. Process. Lett. 99(4), 149–153 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kalyanasundaram, B., Schnitger, G.: The Probabilistic Communication Complexity of Set Intersection. SIAM J. Discrete Math. 5(4), 545–557 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge Univ. Pr. (1997)

    Google Scholar 

  12. Lee, T., Shraibman, A.: Lower Bounds in Communication Complexity. Foundations and Trends in Theoretical Computer Science 3(4), 363–399 (2009)

    MathSciNet  Google Scholar 

  13. Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. Random Structures & Algorithms 34(3), 368–394 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luo, Z.Q., Tsitsiklis, J.N.: On the communication complexity of distributed algebraic computation. J. ACM 40, 1019–1047 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Miltersen, P.B., Nisan, N., Safra, S., Wigderson, A.: On Data Structures and Asymmetric Communication Complexity. J. Comput. Syst. Sci. 57(1), 37–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Munro, J.I., Paterson, M.S.: Selection and sorting with limited storage. Theoretical Computer Science 12(3), 315–323 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Muthukrishnan, S.: Data streams: Algorithms and applications. Now Publishers Inc. (2005)

    Google Scholar 

  18. Newman, I.: Private vs. common random bits in communication complexity. Inf. Process. Lett. 39(2), 67–71 (1991)

    Article  MATH  Google Scholar 

  19. Phillips, J.M., Verbin, E., Zhang, Q.: Lower bounds for number-in-hand multiparty communication complexity, made easy. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 486–501 (2012)

    Google Scholar 

  20. Razborov, A.A.: On the distributional complexity of disjointness. Theoretical Computer Science 106(2), 385–390 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sarlos, T.: Improved approximation algorithms for large matrices via random projections. In: 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 143–152. IEEE (2006)

    Google Scholar 

  22. Sherstov, A.A.: The pattern matrix method for lower bounds on quantum communication. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC 2008, pp. 85–94. ACM, New York (2008)

    Google Scholar 

  23. Sherstov, A.A.: Strong direct product theorems for quantum communication and query complexity. SIAM Journal on Computing 41(5), 1122–1165 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shi, Y., Zhu, Y.: Quantum communication complexity of block-composed functions. Quantum Information and Computation 9, 444–460 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Sun, X., Wang, C.: Randomized Communication Complexity for Linear Algebra Problems over Finite Fields. In: Dürr, C., Wilke, T. (eds.) 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics, LIPIcs (2012)

    Google Scholar 

  26. Sun, X., Wang, C., Yu, W.: The relationship between inner product and counting cycles. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 643–654. Springer, Heidelberg (2012)

    Google Scholar 

  27. Woodruff, D.P., Zhang, Q.: Tight bounds for distributed functional monitoring. In: STOC, pp. 941–960 (2012)

    Google Scholar 

  28. Woodruff, D.P., Zhang, Q.: When distributed computation is communication expensive. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 16–30. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  29. Woodruff, D.P., Zhang, Q.: An optimal lower bound for distinct elements in the message passing model. In: SODA, pp. 718–733 (2014)

    Google Scholar 

  30. Yao, A.C.C.: Some complexity questions related to distributive computing (Preliminary Report). In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, pp. 209–213. ACM (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck Institute for Informatics, Germany

    Yi Li

  2. Institute of Computing Technology, Chinese Academy of Sciences, China

    Xiaoming Sun

  3. Google Inc., USA

    Chengu Wang

  4. IBM Almaden Research Center, USA

    David P. Woodruff

Authors
  1. Yi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoming Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chengu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. David P. Woodruff
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Freiburg, 79110, Freiburg, Germany

    Fabian Kuhn

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Li, Y., Sun, X., Wang, C., Woodruff, D.P. (2014). On the Communication Complexity of Linear Algebraic Problems in the Message Passing Model. In: Kuhn, F. (eds) Distributed Computing. DISC 2014. Lecture Notes in Computer Science, vol 8784. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45174-8_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-45174-8_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-45173-1

  • Online ISBN: 978-3-662-45174-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation