Separating Deterministic from Nondeterministic NOF Multiparty Communication Complexity

(Extended Abstract)
  • Paul Beame
  • Matei David
  • Toniann Pitassi
  • Philipp Woelfel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

We solve some fundamental problems in the number-on-forehead (NOF) k-party communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a one-sided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for k = nO(1) players, where n is the number of bits on each players’ forehead. This separates the analogues of RP and P in the NOF communication model. We also show that there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential and we do not know of any explicit function which allows such separations. However, for the 3-player case we exhibit an explicit function which has Ω(loglogn) randomized complexity for private coins but only constant complexity for public coins.

It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic k-party communication complexity does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the number-in-hand model, is not complete in the NOF model under cylindrical reductions.

References

  1. 1.
    Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory (preliminary version). In: FOCS27th, pp. 337–347 (1986)Google Scholar
  2. 2.
    Babai, L., Gál, A., Kimmel, P.G., Lokam, S.V.: Communication complexity of simultaneous messages. SJCOMP 33, 137–166 (2004)Google Scholar
  3. 3.
    Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. JCSS 45, 204–232 (1992)MATHMathSciNetGoogle Scholar
  4. 4.
    Beame, P., Pitassi, T., Segerlind, N.: Lower bounds for lovász-schrijver systems and beyond follow from multiparty communication complexity. In: ICALP 32nd, pp. 1176–1188 (2005)Google Scholar
  5. 5.
    Beigel, R., Gasarch, W., Glenn, J.: The multiparty communication complexity of Exact-T: Improved bounds and new problems. In: MFCS31st, pp. 146–156 (2006)Google Scholar
  6. 6.
    Beigel, R., Tarui, J.: On ACC. In: FOCS 32nd, pp. 783–792 (1991)Google Scholar
  7. 7.
    Chandra, A.K., Furst, M.L., Lipton, R.J.: Multi-party protocols. In: STOC 15th, pp. 94–99 (1983)Google Scholar
  8. 8.
    Chung, F.R.K., Tetali, P.: Communication complexity and quasi randomness. SIAMDM 6, 110–125 (1993)MATHMathSciNetGoogle Scholar
  9. 9.
    Ford, J., Gál, A.: Hadamard tensors and lower bounds on multiparty communication complexity. In: ICALP 32nd, pp. 1163–1175 (2005)Google Scholar
  10. 10.
    Håstad, J., Goldmann, M.: On the power of small-depth threshold circuits. CompCompl 1, 113–129 (1991)MATHGoogle Scholar
  11. 11.
    Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  12. 12.
    Mansour, Y., Nisan, N., Tiwari, P.: The computational complexity of universal hashing. TCS 107, 121–133 (1993)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Newman, I.: Private vs. common random bits in communication complexity. IPL 39, 67–71 (1991)MATHCrossRefGoogle Scholar
  14. 14.
    Nisan, N.: The communication complexity of threshold gates. In: Proceedings of Combinatorics, Paul Erdos is Eighty, pp. 301–315 (1993)Google Scholar
  15. 15.
    Nisan, N., Wigderson, A.: Rounds in communication complexity revisited. SJCOMP 22, 211–219 (1993)MATHMathSciNetGoogle Scholar
  16. 16.
    Raz, R.: The BNS-Chung criterion for multi-party communication complexity. CompCompl 9, 113–122 (2000)MATHMathSciNetGoogle Scholar
  17. 17.
    Yao, A.C.C.: Some complexity questions related to distributive computing (preliminary report). In: STOC 11th, pp. 209–213 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Paul Beame
    • 1
  • Matei David
    • 2
  • Toniann Pitassi
    • 2
  • Philipp Woelfel
    • 2
  1. 1.University of Washington 
  2. 2.University of Toronto 

Personalised recommendations