Advertisement

Abstract

Gap Hamming Distance is a well-studied problem in communication complexity, in which Alice and Bob have to decide whether the Hamming distance between their respective n-bit inputs is less than \(n/2-\sqrt{n}\) or greater than \(n/2+\sqrt{n}\). We show that every k-round bounded-error communication protocol for this problem sends a message of at least Ω(n/(k 2logk)) bits. This lower bound has an exponentially better dependence on the number of rounds than the previous best bound, due to Brody and Chakrabarti. Our communication lower bound implies strong space lower bounds on algorithms for a number of data stream computations, such as approximating the number of distinct elements in a stream.

Keywords

Communication Complexity Gap Hamming Distance Round Elimination Measure Concentration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ajtai, M.: A lower bound for finding predecessors in Yao’s cell probe model. Combinatorica 8, 235–247 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ball, K.: An elementary introduction to modern convex geometry. Flavors of Geometry 31 (1997)Google Scholar
  3. 3.
    Barvinok, A.: Lecture notes on measure concentration (2005), http://www.math.lsa.umich.edu/~barvinok/total710.pdf
  4. 4.
    Brieden, A., Gritzmann, P., Kannan, R., Klee, V., Lovász, L., Simonovits, M.: Approximation of diameters: Randomization doesn’t help. In: Proceedings of 39th IEEE Symposium on Foundations of Computer Science (FOCS 1998), pp. 244–251 (1998)Google Scholar
  5. 5.
    Brody, J., Chakrabarti, A.: A multi-round communication lower bound for Gap Hamming and some consequences. In: Proceedings of 24th IEEE Conference on Computational Complexity (CCC 2009), pp. 358–368 (2009)Google Scholar
  6. 6.
    Chakrabarti, A., Regev, O.: Tight lower bound for the Gap Hamming problem. Personal Communication (2009)Google Scholar
  7. 7.
    Chvátal, V.: The tail of the hypergeometric distribution. Discrete Mathematics 25(3), 285–287 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Harper, L.: Optimal numbering and isoperimetric problems on graphs. Journal of Combinatorial Theory 1, 385–393 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Indyk, P., Woodruff, D.: Tight lower bounds for the distinct elements problem. In: Proceedings of 44th IEEE Symposium on Foundations of Computer Science (FOCS 2003), pp. 283–289 (2003)Google Scholar
  11. 11.
    Jayram, T.S., Kumar, R., Sivakumar, D.: The one-way communication complexity of Hamming distance. Theory of Computing 4(1), 129–135 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  13. 13.
    Lee, T.,, S.: Disjointness is hard in the multi-party number-on-the-forehead model. In: Proceedings of 23rd IEEE Conference on Computational Complexity (CCC 2008), pp. 81–91 (2008)Google Scholar
  14. 14.
    Lévy, P.: Problèmes concrets d’analyse fonctionnelle. Gauthier-Villars (1951)Google Scholar
  15. 15.
    Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. In: Proceedings of 39th ACM Symposium on the Theory of Computing (STOC 2007), pp. 699–708 (2007)Google Scholar
  16. 16.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  17. 17.
    Miltersen, P., Nisan, N., Safra, S., Wigderson, A.: On data structures and asymmetric communication complexity. J. Comput. Syst. Sci. 57(1), 37–49 (1998); preliminary version in Proceedings of 27th ACM Symposium on the Theory of Computing (STOC 1995), pp. 103–111 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Razborov, A.: Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Science, Mathematics 67, 0204025 (2002)Google Scholar
  19. 19.
    Sherstov, A.: The pattern matrix method for lower bounds on quantum communication. In: Proceedings of 40th ACM Symposium on the Theory of Computing (STOC 2008), pp. 85–94 (2008)Google Scholar
  20. 20.
    Woodruff, D.: Optimal space lower bounds for all frequency moments. In: Proceedings of 15th ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 167–175 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Joshua Brody
    • 1
  • Amit Chakrabarti
    • 1
  • Oded Regev
    • 2
  • Thomas Vidick
    • 3
  • Ronald de Wolf
    • 4
  1. 1.Department of Computer ScienceDartmouth CollegeHanover
  2. 2.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Department of Computer ScienceUCBerkeley
  4. 4.CWI Amsterdam 

Personalised recommendations