On the Role of Shared Randomness in Simultaneous Communication

  • Mohammad Bavarian
  • Dmitry Gavinsky
  • Tsuyoshi Ito
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

Suppose two parties who are interested in performing certain distributed computational tasks are given access to a source of correlated random bits ρ. This source of correlated randomness could be quite useful to the parties for solving various distributed computational problems as it enables the parties to act in a correlated manner. In this work, we initiate the study of power of different sources of shared randomness ρ in the setting of communication complexity; we shall do so in the model of simultaneous message passing (SMP) model of communication complexity, and we shall also argue that this model is the appropriate choice among the commonly studied models of two-party communication complexity for the purpose of studying shared randomness as a resource. As such, we introduce a natural measure for the strength of the correlation provided by a bipartite distribution that we call collision complexity. We demonstrate that the collision complexity col ρ (n) of a bipartite distribution ρ tightly characterises the power of ρ as a resource. We also uncover some surprising phenomenon by showing that even the noisiest shared randomness increases the power of SMP substantially: the equality function can be solved very efficiently with virtually any nontrivial shared randomness— whereas without shared randomness the complexity is known to be \(\Omega(\sqrt n)\).

Keywords

Entropy Expense Bonami 

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References

  1. 1.
    Babai, L., Kimmel, P.G.: Randomized simultaneous messages: Solution of a problem of Yao. In: Communication Complexity (1997)Google Scholar
  2. 2.
    Chor, B., Goldreich, O.: Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing 17, 230–261 (1988)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Gavinsky, D., Ito, T., Wang, G.: Shared randomness and quantum communication in the multi-party model. In: IEEE 28th Conference on Computational Complexity (CCC), pp. 34–43 (2013)Google Scholar
  4. 4.
    Gavinsky, D., Kempe, J., Regev, O., de Wolf, R.: Bounded-error quantum state identification and exponential separations in communication complexity. SIAM Journal on Computing 39, 1–24 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press (1997)Google Scholar
  6. 6.
    Mossel, E., O’Donnell, R.: Coin flipping from a cosmic source: On error correction of truly random bits. Random Structures and Algorithms 26, 418–436 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Mossel, E., O’Donnell, R., Regev, O., Steif, J.E., Sudakov, B.: Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami–Beckner inequality. Israel Journal of Mathematics 154, 299–336 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Newman, I.: Private vs. common random bits in communication complexity. Information Processing Letters 39, 67–71 (1991)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Newman, I., Szegedy, M.: Public vs. private coin flips in one round communication games. In: Proceedings of the 28th Symposium on Theory of Computing, pp. 561–570 (1996)Google Scholar
  10. 10.
    Witsenhausen, H.S.: On sequences of pairs of dependent random variables. SIAM Journal on Applied Mathematics 28, 100–113 (1975)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Yang, K.: On the (Im)possibility of Non-interactive Correlation Distillation, vol. 382 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mohammad Bavarian
    • 1
  • Dmitry Gavinsky
    • 2
  • Tsuyoshi Ito
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeU.S.A.
  2. 2.Institute of MathematicsAcademy of SciencesPrahaCzech Republic

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