Abstract
We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the “Number on the Forehead” model of Chandra, Furst, and Lipton. We define an analog of the Hadamard property of matrices for tensors in multiple dimensions and show that any k-party communication problem represented by a Hadamard tensor must have Ω(n/2k) multiparty communication complexity. We also exhibit constructions of Hadamard tensors. This allows us to obtain Ω(n/2k) lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions.
Similar content being viewed by others
References
N. Alon & J. H. Spencer (2000). The Probabilistic Method (second edition). John Willey & Sons.
L. Babai, T. P. Hayes & P. G. Kimmel (1998). The Cost of the Missing Bit: Communication Complexity with Help. Proceedings of the 30th Annual ACM Symposium on Theory of Computing 673–682.
Babai L., Nisan N., Szegedy M. (1992) Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs. Journal of Computer and System Sciences 45(2): 204–232
Barrington D. (1989) Bounded-width polynomial size branching programs recognize exactly those languages in NC 1. Journal of Computer and System Sciences 38(1): 150–164
P. Beame & D. Huynh-Ngoc (2009). Multiparty communication complexity and threshold circuit size of AC 0. Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science 53–62.
R. Beigel & J. Tarui (1991). On ACC. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science 783–792.
Best M.R. (1977) The Excess of a Hadamard Matrix. Indag. Math. 39(5): 357–361
A. Chandra, M. Furst & R. Lipton (1983). Multiparty protocols. Proceedings of the 15th Annual ACM Symposium on Theory of Computing 94–99.
A. Chattopadhyay (2007). Discrepancy and the power of bottom fan-in depth-three circuits. Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science 449–458.
A. Chattopadhyay & A. Ada (2008). Multiparty communication complexity of disjointness, Technical report TR-08-002. Electronic Colloquium on Computational Complexity.
Chor B., Goldreich O. (1988) Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity. SIAM J. Computing 17: 230–261
Chung F. (1990) Quasi-Random Classes of Hypergraphs. Random Structures and Algorithms 1(4): 363–382
Chung F., Tetali P. (1993) Communication complexity and quasi randomness. SIAM J. Discrete Math. 6(1): 110–123
Erdős P., Spencer J.H. (1974) Probabilistic methods in combinatorics. Academic Press, New York
Grolmusz V. (1994) The BNS Lower Bound for Multi-Party Protocols is Nearly Optimal. Information and Computation 112: 51–54
J. Håstad & M. Goldmann (1991). On the power of small depth threshold circuits. Computational Complexity 113–129.
Kharaghani H., Tayfez-Rezaie B. (2005) A Hadamard matrix of order 428. Journal of Combinatorial Designs 13: 435–440
Klauck H. (2007) Lower bounds for quantum communication complexity. SIAM J. Comput. 37(1): 20–46
E. Kushilevitz & N. Nisan (1997). Communication complexity. Cambridge University Press.
Lee T., Shraibman A. (2009) Disjointness is hard in the multi-party number-on-the-forehead model. Computational Complexity 18(2): 309–336
J. H. van Lint & R. M. Wilson (1992). A course in combinatorics. Cambridge University Press.
J.H. van Lint (1991). Introduction to Coding Theory. Springer-Verlag.
Nisan N., Wigderson A. (1993) Rounds in communication complexity revisited. SIAM J. Comp. 22: 211–219
P. Pudlák (1994). Unexpected upper bounds on the complexity of some communication games. Proceedings of the 21st International Colloquium on Automata, Languages and Programming 1–11.
Pudlák P., Rödl V., Sgall J. (1997) Boolean circuits, tensor ranks and communication complexity. SIAM J. Comp. 26: 605–633
Raz R. (2000) The BNS-Chung criterion for multiparty communication complexity. Computational Complexity 9(2): 113–122
W. M. Schmidt (1978). Equations over finite fields: an elementary approach. Lecture Notes in Math., Springer Verlag 536.
Shoup V. (1990) New Algorithms for Finding Irreducible Polynomials over Finite Fields. Mathematics of Computation 54: 435–447
J. Spencer (1987). Ten lectures on the probabilistic method. Society for Industrial and Applied Mathematics.
Weil A. (1948) On some exponential sums. Proceedings of the National Academy of Sciences of the United States of America 34(5): 204–207
R. Williams (2011). Non-Uniform ACC Circuit Lower Bounds. To appear in Proc. of the 26th IEEE Conference on Computational Complexity.
A. Yao (1979). Some complexity questions related to distributed computing. Proceedings of the 11th Annual ACM Symposium on Theory of Computing 209–213.
A. Yao (1990). On ACC and threshold circuits. Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science 619–627.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ford, J., Gál, A. Hadamard tensors and lower bounds on multiparty communication complexity. comput. complex. 22, 595–622 (2013). https://doi.org/10.1007/s00037-012-0052-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00037-012-0052-6