skip to main content
research-article

Inner Product and Set Disjointness: Beyond Logarithmically Many Parties

Authors Info & Claims
Published:25 November 2020Publication History
Skip Abstract Section

Abstract

A major goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems f:({0, 1} n)k → {0, 1} with k > log n parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every k ≥ log n, showing in both cases that Θ(1 + ⌈log n⌉/ log ⌈1 + k/ log n⌉) bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is knϵ for some constant ϵ > 0.

References

  1. Anil Ada, Arkadev Chattopadhyay, Omar Fawzi, and Phuong Nguyen. 2015. The NOF multiparty communication complexity of composed functions. Comput. Complex. 24, 3 (2015), 645--694. DOI:https://doi.org/10.1007/s00037-013-0078-4Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. László Babai, Anna Gál, Peter G. Kimmel, and Satyanarayana V. Lokam. 2003. Communication complexity of simultaneous messages. SIAM J. Comput. 33, 1 (2003), 137--166. DOI:https://doi.org/10.1137/S0097539700375944Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. László Babai, Noam Nisan, and Mario Szegedy. 1992. Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci. 45, 2 (1992), 204--232. DOI:https://doi.org/10.1016/0022-0000(92)90047-MGoogle ScholarGoogle ScholarDigital LibraryDigital Library
  4. Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. 2004. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68, 4 (2004), 702--732. DOI:https://doi.org/10.1016/j.jcss.2003.11.006Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Paul Beame and Trinh Huynh. 2012. Multiparty communication complexity and threshold circuit size of AC0. SIAM J. Comput. 41, 3 (2012), 484--518. DOI:https://doi.org/10.1137/100792779Google ScholarGoogle ScholarCross RefCross Ref
  6. Paul Beame, Toniann Pitassi, and Nathan Segerlind. 2007. Lower bounds for lovász-schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput. 37, 3 (2007), 845--869. DOI:https://doi.org/10.1137/060654645Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Paul Beame, Toniann Pitassi, Nathan Segerlind, and Avi Wigderson. 2006. A strong direct product theorem for corruption and the multiparty communication complexity of disjointness. Comput. Complex. 15, 4 (2006), 391--432. DOI:https://doi.org/10.1007/s00037-007-0220-2Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Ashok K. Chandra, Merrick L. Furst, and Richard J. Lipton. 1983. Multi-party protocols. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC’83). 94--99. DOI:https://doi.org/10.1145/800061.808737Google ScholarGoogle Scholar
  9. Arkadev Chattopadhyay and Anil Ada. 2008. Multiparty communication complexity of disjointness. In Proceedings of the Electronic Colloquium on Computational Complexity (ECCC’08). Report TR08-002.Google ScholarGoogle Scholar
  10. Arkadev Chattopadhyay and Michael E. Saks. 2014. The power of super-logarithmic number of players. In Proceedings of the 18th International Workshop on Randomization and Computation (RANDOM’14). 596--603. DOI:https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.596Google ScholarGoogle Scholar
  11. Benny Chor and Oded Goldreich. 1988. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 2 (1988), 230--261. DOI:https://doi.org/10.1137/0217015Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Fan R. K. Chung and Prasad Tetali. 1993. Communication complexity and quasi randomness. SIAM J. Discrete Math. 6, 1 (1993), 110--123. DOI:https://doi.org/10.1137/0406009Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Jeffrey Stephen Ford. 2006. Lower Bound Methods for Multiparty Communication Complexity. Ph.D. Dissertation. The University of Texas at Austin.Google ScholarGoogle Scholar
  14. Vince Grolmusz. 1994. The BNS lower bound for multi-party protocols in nearly optimal. Info. Comput. 112, 1 (1994), 51--54. DOI:https://doi.org/10.1006/inco.1994.1051Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Johan Håstad and Mikael Goldmann. 1991. On the power of small-depth threshold circuits. Comput. Complex. 1 (1991), 113--129. DOI:https://doi.org/10.1007/BF01272517Google ScholarGoogle ScholarCross RefCross Ref
  16. Stasys Jukna. 2001. Extremal Combinatorics with Applications in Computer Science. Springer-Verlag, Berlin. DOI:https://doi.org/10.1007/978-3-662-04650-0Google ScholarGoogle Scholar
  17. Bala Kalyanasundaram and Georg Schnitger. 1992. The probabilistic communication complexity of set intersection. SIAM J. Discrete Math. 5, 4 (1992), 545--557. DOI:https://doi.org/10.1137/0405044Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Eyal Kushilevitz and Noam Nisan. 1997. Communication Complexity. Cambridge University Press.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Troy Lee and Adi Shraibman. 2009. Disjointness is hard in the multiparty number-on-the-forehead model. Comput. Complex. 18, 2 (2009), 309--336. DOI:https://doi.org/10.1007/s00037-009-0276-2Google ScholarGoogle ScholarCross RefCross Ref
  20. Noam Nisan and Mario Szegedy. 1994. On the degree of boolean functions as real polynomials. Comput. Complex. 4 (1994), 301--313. DOI:https://doi.org/10.1007/BF01263419Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Ran Raz. 2000. The BNS-chung criterion for multi-party communication complexity. Comput. Complex. 9, 2 (2000), 113--122. DOI:https://doi.org/10.1007/PL00001602Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Alexander A. Razborov. 1992. On the distributional complexity of disjointness. Theor. Comput. Sci. 106, 2 (1992), 385--390. DOI:https://doi.org/10.1016/0304-3975(92)90260-MGoogle ScholarGoogle ScholarDigital LibraryDigital Library
  23. Alexander A. Razborov and Avi Wigderson. 1993. nΩ(log n) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. Info. Process. Lett. 45, 6 (1993), 303--307. DOI:https://doi.org/10.1016/0020-0190(93)90041-7Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Alexander A. Sherstov. 2014. Communication lower bounds using directional derivatives. J. ACM 61, 6 (2014), 1--71. DOI:https://doi.org/10.1145/2629334 Preliminary version in Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC’13).Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Alexander A. Sherstov. 2016. The multiparty communication complexity of set disjointness. SIAM J. Comput. 45, 4 (2016), 1450--1489. DOI:https://doi.org/10.1137/120891587 Preliminary version in Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC’09).Google ScholarGoogle ScholarCross RefCross Ref
  26. Pascal Tesson. 2003. Computational Complexity Questions Related to Finite Monoids and Semigroups. Ph.D. Dissertation. McGill University.Google ScholarGoogle Scholar
  27. Andrew Chi-Chih Yao. 1990. On ACC and threshold circuits. In Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science (FOCS’90). 619--627. DOI:https://doi.org/10.1109/FSCS.1990.89583Google ScholarGoogle Scholar

Index Terms

  1. Inner Product and Set Disjointness: Beyond Logarithmically Many Parties

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    • Article Metrics

      • Downloads (Last 12 months)20
      • Downloads (Last 6 weeks)2

      Other Metrics

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    HTML Format

    View this article in HTML Format .

    View HTML Format
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!