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 k ≥ nϵ for some constant ϵ > 0.
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Jeffrey Stephen Ford. 2006. Lower Bound Methods for Multiparty Communication Complexity. Ph.D. Dissertation. The University of Texas at Austin.Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- Stasys Jukna. 2001. Extremal Combinatorics with Applications in Computer Science. Springer-Verlag, Berlin. DOI:https://doi.org/10.1007/978-3-662-04650-0Google Scholar
- 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 Scholar
Digital Library
- Eyal Kushilevitz and Noam Nisan. 1997. Communication Complexity. Cambridge University Press.Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Cross Ref
- Pascal Tesson. 2003. Computational Complexity Questions Related to Finite Monoids and Semigroups. Ph.D. Dissertation. McGill University.Google Scholar
- 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 Scholar
Index Terms
Inner Product and Set Disjointness: Beyond Logarithmically Many Parties
Recommendations
Simplified lower bounds on the multiparty communication complexity of disjointness
CCC '15: Proceedings of the 30th Conference on Computational ComplexityWe show that the deterministic number-on-forehead communication complexity of set disjointness for k parties on a universe of size n is Ω(n/ 4k). This gives the first lower bound that is linear in n, nearly matching Grolmusz's upper bound of O(log2(n) + ...
On the Communication Complexity of Sparse Set Disjointness and Exists-Equal Problems
FOCS '13: Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer ScienceIn this paper we study the two player randomized communication complexity of the sparse set disjoint ness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these ...
The Communication Complexity of Set Intersection and Multiple Equality Testing
In this paper we explore fundamental problems in randomized communication complexity such as computing SetIntersection on sets of size $k$ and EqualityTesting between vectors of length $k$. Sağlam and Tardos [Proceedings of the 54th Annual IEEE Symposium on ...






Comments