Improved Separations between Nondeterministic and Randomized Multiparty Communication

  • Matei David
  • Toniann Pitassi
  • Emanuele Viola
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5171)

Abstract

We exhibit an explicit function f : {0, 1}n →{0,1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(logn) bits, but that requires nΩ(1) bits of communication for randomized number-on-forehead protocols with k = δ·logn players, for any fixed δ< 1. Recent breakthrough results for the Set-Disjointness function (Sherstov, STOC ’08; Lee Shraibman, CCC ’08; Chattopadhyay Ada, ECCC ’08) imply such a separation but only when the number of players is k < loglogn.

We also show that for any k = A loglogn the above function f is computable by a small circuit whose depth is constant whenever A is a (possibly large) constant. Recent results again give such functions but only when the number of players is k < loglogn.

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References

  1. [ABI86]
    Alon, N., Babai, L., Itai, A.: A fast and simple randomized algorithm for the maximal independent set problem. Journal of Algorithms 7, 567–583 (1986)MATHCrossRefMathSciNetGoogle Scholar
  2. [BDPW07]
    Beame, P., David, M., Pitassi, T., Woelfel, P.: Separating deterministic from nondet, nof multiparty communication complexity. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 134–145. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. [BFS86]
    Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory (preliminary version). In: FOCS, pp. 337–347. IEEE, Los Alamitos (1986)Google Scholar
  4. [BNS92]
    Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. System Sci. 45(2), 204–232 (1992)MATHCrossRefMathSciNetGoogle Scholar
  5. [Bop97]
    Boppana, R.B.: The average sensitivity of bounded-depth circuits. Inform. Process. Lett. 63(5), 257–261 (1997)CrossRefMathSciNetGoogle Scholar
  6. [BPS07]
    Beame, P., Pitassi, T., Segerlind, N.: Lower bounds for lovász–schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput. 37(3), 845–869 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. [CA08]
    Chattopadhyay, A., Ada, A.: Multiparty communication complexity of disjointness. ECCC, Technical Report TR08-002 (2008)Google Scholar
  8. [CFL83]
    Chandra, A.K., Furst, M.L., Lipton, R.J.: Multi-party protocols. In: STOC (1983)Google Scholar
  9. [CG89]
    Chor, B., Goldreich, O.: On the power of two-point based sampling. Journal of Complexity 5(1), 96–106 (1989)MATHCrossRefMathSciNetGoogle Scholar
  10. [Cha07]
    Chattopadhyay, A.: Discrepancy and the power of bottom fan-in in depth-three circuits. In: FOCS, October 2007, pp. 449–458. IEEE, Los Alamitos (2007)Google Scholar
  11. [CT93]
    Chung, F.R.K., Tetali, P.: Communication complexity and quasi randomness. SIAM J. Discrete Math. 6(1), 110–123 (1993)MATHCrossRefMathSciNetGoogle Scholar
  12. [FG05]
    Ford, J., Gál, A.: Hadamard tensors and lower bounds on multiparty communication complexity. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1163–1175. Springer, Heidelberg (2005)Google Scholar
  13. [GV04]
    Gutfreund, D., Viola, E.: Fooling parity tests with parity gates. In: Jansen, K., Khanna, S., Rolim, J., Ron, D. (eds.) RANDOM 2004. LNCS, vol. 3122, pp. 381–392. Springer, Heidelberg (2004)Google Scholar
  14. [HG91]
    Håstad, J., Goldmann, M.: On the power of small-depth threshold circuits. Comput. Complexity 1(2), 113–129 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. [HV06]
    Healy, A., Viola, E.: Constant-depth circuits for arithmetic in finite fields of characteristic two. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 672–683. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. [KN97]
    Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  17. [LMN93]
    Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40(3), 607–620 (1993)MATHMathSciNetGoogle Scholar
  18. [LS08]
    Lee, T., Shraibman, A.: Disjointness is hard in the multi-party number on the forehead model. In: CCC. IEEE, Los Alamitos (2008)Google Scholar
  19. [MNT90]
    Mansour, Y., Nisan, N., Tiwari, P.: The computational complexity of universal hashing. In: STOC, pp. 235–243. ACM Press, New York (1990)Google Scholar
  20. [NN93]
    Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993)MATHCrossRefMathSciNetGoogle Scholar
  21. [NS94]
    Nisan, N., Szegedy, M.: On the degree of boolean functions as real polynomials. Computational Complexity 4, 301–313 (1994)MATHCrossRefMathSciNetGoogle Scholar
  22. [NW93]
    Nisan, N., Wigderson, A.: Rounds in communication complexity revisited. SIAM J. Comput. 22(1), 211–219 (1993)MATHCrossRefMathSciNetGoogle Scholar
  23. [Pat92]
    Paturi, R.: On the degree of polynomials that approximate symmetric boolean functions (preliminary version). In: STOC, pp. 468–474. ACM, New York (1992)Google Scholar
  24. [Raz87]
    Razborov, A.A.: Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Mat. Zametki 41(4), 598–607, 623 (1987)MathSciNetGoogle Scholar
  25. [Raz00]
    Raz, R.: The BNS-Chung criterion for multi-party communication complexity. Comput. Complexity 9(2), 113–122 (2000)MATHCrossRefMathSciNetGoogle Scholar
  26. [Raz03]
    Razborov, A.: Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics 67(1), 145–159 (2003)MATHCrossRefMathSciNetGoogle Scholar
  27. [RW93]
    Razborov, A., Wigderson, A.: \(n\sp {\Omega(\log n)}\) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. Inform. Process. Lett. 45(6), 303–307 (1993)MATHCrossRefMathSciNetGoogle Scholar
  28. [She07]
    Sherstov, A.: Separating AC0 from depth-2 majority circuits. In: STOC 2007: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (2007)Google Scholar
  29. [She08a]
    Sherstov, A.: The pattern matrix method for lower bounds on quantum communication. In: STOC (2008)Google Scholar
  30. [She08b]
    Sherstov, A.A.: Communication lower bounds using dual polynomials. Electronic Colloquium on Computational Complexity, Technical Report TR08-057 (2008)Google Scholar
  31. [VW07]
    Viola, E., Wigderson, A.: Norms, xor lemmas, and lower bounds for GF(2) polynomials and multiparty protocols. In: CCC. IEEE, Los Alamitos (2007); Theory of Computing (to appear) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Matei David
    • 1
  • Toniann Pitassi
    • 1
  • Emanuele Viola
    • 2
  1. 1.Department of Computer ScienceUniversity of Toronto 
  2. 2.Computer Science DepartmentColumbia University 

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