Suppose each of k≤n o(1) players holds an n-bit number x i in its hand. The players wish to determine if ∑ i≤k x i =s. We give a public-coin protocol with error 1% and communication O(k logk). The communication bound is independent of n, and for k≥3 improves on the O(k logn) bound by Nisan (Bolyai Soc. Math. Studies; 1993).
Our protocol also applies to addition modulo m. In this case we give a matching (public-coin) Ω(k logk) lower bound for various m. We also obtain some lower bounds over the integers, including Ω (k log logk) for protocols that are one-way, like ours.
We give a protocol to determine if ∑x i > s with error 1% and communication O(k logk) log n. For k≥3 this improves on Nisan’s O(k log2 n) bound. A similar improvement holds for computing degree-(k−1) polynomial-threshold functions in the number-on-forehead model.
We give a (public-coin, 2-player, tight) Ω(logn) lower bound to determine if x 1 > x 2. This improves on the Ω(√logn) bound by Smirnov (1988). Troy Lee informed us in January 2013 that an Ω(logn) lower bound may also be obtained by combining a result in learning theory by Forster et al. (2003) with a result by Linial and Shraibman (2009).
As an application, we show that polynomial-size AC0 circuits augmented with O(1) threshold (or symmetric) gates cannot compute cryptographic pseudorandom functions, extending the result about AC0 by Linial, Mansour, and Nisan (J. ACM; 1993).
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J. Aspnes, R. Beigel, M. Furst and S. Rudich: The expressive power of voting polynomials, Combinatorica 14 (1994), 135–148.
N. Alon, O. Goldreich, J. Håstad and R. Peralta: Simple constructions of almost k-wise independent random variables, Random Structures & Algorithms 3 (1992), 289–304.
P. Beame: A switching lemma primer, Technical Report UW-CSE-95-07-01, Depart-ment of Computer Science and Engineering, University of Washington, November 1994, Available from http://www.cs.washington.edu/homes/beame/.
R. Beigel: When do extra majority gates help? polylog(-N) majority gates are equivalent to one, Comput. Complexity 4 (1994), 314–324.
L. Babai, N. Nisan and M. Szegedy: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs, J. of Computer and System Sciences 45 (1992), 204–232.
M. Ben-Or and A. Hassidim: The bayesian learner is optimal for noisy binary search (and pretty good for quantum as well), in: IEEE Symp. on Foundations of Computer Science (FOCS), 221–230, 2008.
R. Boppana and M. Sipser: The complexity of finite functions, in: Handbook of theoretical computer science, Vol. A, 757–804. Elsevier, Amsterdam, 1990.
A. Bogdanov and E. Viola: Pseudorandom bits for polynomials, SIAM J. on Computing 39 (2010), 2464–2486.
A. Baker and G. Wüstholz: Logarithmic forms and Diophantine geometry, volume 9 of New Mathematical Monographs, Cambridge University Press, 2007.
M. Braverman and O. Weinstein: A discrepancy lower bound for information complexity, in: Workshop on Randomization and Computation (RANDOM), 459–470, 2012.
A. K. Chandra, M. L. Furst and R. J. Lipton: Multi-party protocols, in: 15th ACM Symp. on the Theory of Computing (STOC), 94–99, 1983.
I. Csiszár and J. Körner: Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, Inc., 1982.
A. Cobham: The intrinsic computational diffculty of functions, in: Int. Congress for Logic, Methodology and Philosophy of Science, 24–30, 1964.
F. R. K. Chung and P. Tetali: Communication complexity and quasi randomness, SIAM J. Discrete Math. 6 (1993), 110–123.
M. Dietzfelbinger, T. Hagerup, J. Katajainen and M. Penttonen: A reliable randomized algorithm for the closest-pair problem, J. Algorithms 25 (1997), 19–51.
C. Dutta, G. Pandurangan, R. Rajaraman, Z. Sun and E. Viola: On the complexity of information spreading in dynamic networks, in: ACM-SIAM Symp. on Discrete Algorithms (SODA), 2013.
U. Feige, P. Raghavan, D. Peleg and E. Upfal: Computing with noisy information, SIAM J. Comput. 23 (1994), 1001–1018.
M. L. Furst, J. B. Saxe and M. Sipser: Parity, circuits, and the polynomial-time hierarchy, Mathematical Systems Theory 17 (1984), 13–27.
J. Forster, N. Schmitt, H.-U. Simon and T. Suttorp: Estimating the optimal margins of embeddings in euclidean half spaces, Machine Learning 51 (2003), 263–281.
O. Goldreich, S. Goldwasser and S. Micali: How to construct random functions, J. of the ACM 33 (1986), 792–807.
M. Goldmann, J. Håstad and A. A. Razborov: Majority gates vs. general weighted threshold gates, Computational Complexity 2 (1992), 277–300.
P. Gopalan and R. A. Servedio: Learning and lower bounds for AC0 with threshold gates, in: Workshop on Randomization and Computation (RANDOM), 588–601, 2010.
J. Håstad: Computational limitations of small-depth circuits, MIT Press, 1987.
J. Håstad: On the size of weights for threshold gates, SIAM J. Discrete Math. 7 (1994), 484–492.
J. Håstad and M. Goldmann: On the power of small-depth threshold circuits, Comput. Complexity 1 (1991), 113–129.
M. Krause and S. Lucks: On the minimal hardware complexity of pseudorandom function generators, in: Symp. on Theoretical Aspects of Computer Science (STACS), 419–430, 2001.
E. Kushilevitz and N. Nisan: Communication complexity, Cambridge University Press, 1997.
L. A. Levin: One way functions and pseudorandom generators, Combinatorica 7 (1987), 357–363.
N. Linial, Y. Mansour and N. Nisan: Constant depth circuits, Fourier transform, and learnability, J. of the ACM 40 (1993), 607–620.
N. Linial and A. Shraibman: Learning complexity vs communication complexity, Combinatorics, Probability & Computing 18 (2009), 227–245.
P. Bro Miltersen, N. Nisan, S. Safra and A. Wigderson: On data structures and asymmetric communication complexity, J. of Computer and System Sciences, 57 (1998), 37–49.
S. Muroga, I. Toda and S. Takasu: Theory of majority decision elements, J. Franklin Inst. 271 (1961), 376–418.
S. Muroga: Threshold logic and its applications, Wiley-Interscience, New York, 1971.
V. A. Nepomnjaščiĭ: Rudimentary predicates and Turing calculations, Soviet Mathematics-Doklady 11 (1970), 1462–1465.
N. Nisan: The communication complexity of threshold gates, in: Combinatorics, Paul Erdős is Eighty, number 1 in Bolyai Society Mathematical Studies, 301–315, 1993.
J. Naor and M. Naor: Small-bias probability spaces: effcient constructions and applications, SIAM J. on Computing 22 (1993), 838–856.
M. Naor and O. Reingold: Synthesizers and their application to the parallel construction of pseudo-random functions, J. Comput. Syst. Sci. 58 (1999), 336–375.
M. Naor and O. Reingold: Number-theoretic constructions of efficient pseudorandom functions, J. of the ACM 51 (2004), 231–262.
M. Naor, O. Reingold and A. Rosen: Pseudorandom functions and factoring, SIAM J. Comput. 31 (2002), 1383–1404.
N. Nisan and D. Zuckerman: Randomness is linear in space, J. of Computer and System Sciences 52 (1996), 43–52.
V. Podol’skiĭ: Perceptrons of large weight, Problems of Information Transmission 45 (2009), 46–53.
R. Raz: The BNS-Chung criterion for multi-party communication complexity, Comput. Complexity 9 (2000), 113–122.
A. Razborov and S. Rudich: Natural proofs, J. of Computer and System Sciences 55 (1997), 24–35.
D. V. Smirnov: Shannon’s information methods for lower bounds for probabilistic communication complexity, Master’s thesis, Moscow University, 1988.
H. Straubing and D. Théerien: A note on modp — modm circuits, Theory Comput. Syst. 39 (2006), 699–706.
R. Shaltiel and E. Viola: Hardness amplification proofs require majority, SIAM J. on Computing 39 (2010), 3122–3154.
E. Viola: Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates, SIAM J. on Computing 36 (2007), 1387–1403.
E. Viola: Cell-probe lower bounds for prefix sums, 2009, arXiv:0906.1370v1.
E. Viola and A. Wigderson: Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols, Theory of Computing 4 (2008), 137–168.
Supported by NSF grants CCF-0845003, CCF-1319206.
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Viola, E. The communication complexity of addition. Combinatorica 35, 703–747 (2015). https://doi.org/10.1007/s00493-014-3078-3
Mathematics Subject Classification (2000)