Fooling Pairs in Randomized Communication Complexity

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9988)


The fooling pairs method is one of the standard methods for proving lower bounds for deterministic two-player communication complexity. We study fooling pairs in the context of randomized communication complexity. We show that every fooling pair induces far away distributions on transcripts of private-coin protocols. We use the above to conclude that the private-coin randomized \(\varepsilon \)-error communication complexity of a function f with a fooling set \(\mathcal S\) is at least order \(\log \frac{\log |\mathcal S|}{\varepsilon }\). This relationship was earlier known to hold only for constant values of \(\varepsilon \). The bound we prove is tight, for example, for the equality and greater-than functions.

As an application, we exhibit the following dichotomy: for every boolean function f and integer n, the (1/3)-error public-coin randomized communication complexity of the function \(\bigvee _{i=1}^{n}f(x_i,y_i)\) is either at most c or at least n/c, where \(c>0\) is a universal constant.


  1. 1.
    Alon, N.: Perturbed identity matrices have high rank: proof and applications. Comb. Probab. Comput. 18(1–2), 3–15 (2009). MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. In: FOCS, pp. 209–218 (2002)Google Scholar
  3. 3.
    Barak, B., Braverman, M., Chen, X., Rao, A.: How to compress interactive communication. SIAM J. Comput. 42(3), 1327–1363 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chor, B., Kushilevitz, E.: A zero-one law for Boolean privacy. SIAM J. Discrete Math. 4(1), 36–47 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hastad, J., Wigderson, A.: The randomized communication complexity of set disjointness. Theor. Comput. 3(1), 211–219 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discrete Math. 5(4), 545–557 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Krause, M.: Geometric arguments yield better bounds for threshold circuits and distributed computing. Theor. Comput. Sci. 156(1–2), 99–117 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, New York (1997)CrossRefMATHGoogle Scholar
  9. 9.
    Lee, T., Shraibman, A.: Lower bounds in communication complexity. Founda. Trends Theoret. Comput. Sci. 3(4), 263–398 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Newman, I.: Private vs. common random bits in communication complexity. Inf. Process. Lett. 39(2), 67–71 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Paturi, R., Simon, J.: Probabilistic communication complexity. J. Comput. Syst. Sci. 33(1), 106–123 (1986)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Yao, A.C.C.: Some complexity questions related to distributive computing (preliminary report). In: STOC, pp. 209–213 (1979)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Technion, Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Computer Science and EngineeringUniversity of WashingtonSeattleUSA
  3. 3.Microsoft ResearchHerzliyaIsrael
  4. 4.Max Planck Institute for InformaticsSaarbrückenGermany

Personalised recommendations