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Improved Separations between Nondeterministic and Randomized Multiparty Communication

Published:01 September 2009Publication History
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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 [Lee and Shraibman 2008; Chattopadhyay and Ada 2008] based on the work of Sherstov [2009; 2008a] 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.

References

  1. Alon, N., Babai, L., and Itai, A. 1986. A fast and simple randomized algorithm for the maximal independent set problem. J. Algo. 7, 567--583. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Babai, L., Frankl, P., and Simon, J. 1986. Complexity classes in communication complexity theory. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 337--347. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Babai, L., Nisan, N., and Szegedy, M. 1992. Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci. 45, 2, 204--232. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Beame, P., David, M., Pitassi, T., and Woelfel, P. 2007. Separating deterministic from nondeterministic not multiparty communication complexity. In Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP). Springer, 134--145. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Beame, P. and Huynh-Ngoc, D.-T. 2008a. Multiparty communication complexity and threshold size of AC 0. Manuscript. http://www.cs.washington.edu/homes/beame/papers/multiac0.pdf.Google ScholarGoogle Scholar
  6. Beame, P. and Huynh-Ngoc, D.-T. 2008b. Multiparty communication complexity of AC 0. Tech. rep. TR08-061, Electronic Colloquium on Computational Complexity. www.eccc.uni-trier.de/.Google ScholarGoogle Scholar
  7. Beame, P., Pitassi, T., and Segerlind, N. 2007. Lower bounds for lovász--schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput. 37, 3, 845--869. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Boppana, R. B. 1997. The average sensitivity of bounded-depth circuits. Inform. Process. Lett. 63, 5, 257--261. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Chandra, A. K., Furst, M. L., and Lipton, R. J. 1983. Multi-party protocols. In Proceedings of the 15th Annual Symposium on Theory of Computing (STOC). 94--99. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Chattopadhyay, A. 2007. Discrepancy and the power of bottom fan-in in depth-three circuits. In Proceedings of the 48th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 449--458. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Chattopadhyay, A. and Ada, A. 2008. Multiparty communication complexity of disjointness. Tech. rep. TR08-002, Electronic Colloquium on Computational Complexity.Google ScholarGoogle Scholar
  12. Chor, B. and Goldreich, O. 1989. On the power of two-point based sampling. J. Complex. 5, 1, 96--106. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Chung, F. R. K. and Tetali, P. 1993. Communication complexity and quasi randomness. SIAM J. Discrete Math. 6, 1, 110--123. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Ford, J. and Gál, A. 2005. Hadamard tensors and lower bounds on multiparty communication complexity. In Proceedings of the 32th International Colloquium on Automata, Languages and Programming (ICALP). Springer, 1163--1175. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Gutfreund, D. and Viola, E. 2004. Fooling parity tests with parity gates. In Proceedings of the 8th International Workshop on Randomization and Computation (RANDOM). Springer, 381--392.Google ScholarGoogle Scholar
  16. Håstad, J. and Goldmann, M. 1991. On the power of small-depth threshold circuits. Comput. Complex. 1, 2, 113--129.Google ScholarGoogle ScholarCross RefCross Ref
  17. Healy, A. and Viola, E. 2006. Constant-depth circuits for arithmetic in finite fields of characteristic two. In Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS). Springer, 672--683. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Kushilevitz, E. and Nisan, N. 1997. Communication Complexity. Cambridge University Press, Cambridge, UK. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Lee, T. and Shraibman, A. 2008. Disjointness is hard in the multi-party number on the forehead model. In Proceedings of the 23nd Annual Conference on Computational Complexity. IEEE. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Linial, N., Mansour, Y., and Nisan, N. 1993. Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40, 3, 607--620. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Mansour, Y., Nisan, N., and Tiwari, P. 1993. The computational complexity of universal hashing. Theor. Comput. Sci. 107, 121--133. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Naor, J. and Naor, M. 1993. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput. 22, 4, 838--856. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Nisan, N. and Szegedy, M. 1994. On the degree of Boolean functions as real polynomials. Computat. Complex. 4, 301--313. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Nisan, N. and Wigderson, A. 1993. Rounds in communication complexity revisited. SIAM J. Comput. 22, 1, 211--219. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Paturi, R. 1992. On the degree of polynomials that approximate symmetric Boolean functions. In Proceedings of the 24th Annual Symposium on Theory of Computing (STOC). ACM, 468--474. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Raz, R. 2000. The BNS-Chung criterion for multi-party communication complexity. Comput. Complex. 9, 2, 113--122. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Razborov, A. 2003. Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics 67, 1, 145--159.Google ScholarGoogle Scholar
  28. Razborov, A. and Wigderson, A. 1993. n 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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Sherstov, A. 2008a. The pattern matrix method for lower bounds on quantum communication. In Proceedings of the 40th Annual Symposium on the Theory of Computing (STOC). ACM, 85--94. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. Sherstov, A. 2008b. Communication lower bounds using dual polynomials. Bull. EATCS 95, 59--93.Google ScholarGoogle Scholar
  31. Sherstov, A. 2009. Separating AC 0 from depth-2 majority circuits. SIAM J. Comput. 38, 6, 2113--2129. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Viola, E. and Wigderson, A. 2008. Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols. Theor. Comput. 4, 137--168.Google ScholarGoogle ScholarCross RefCross Ref

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