Skip to main content
SpringerLink
Log in

The communication complexity of addition

  • Original Paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Aspnes, R. Beigel, M. Furst and S. Rudich: The expressive power of voting polynomials, Combinatorica 14 (1994), 135–148.

    Article  MATH  MathSciNet  Google Scholar 

  2. 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.

    Article  MATH  MathSciNet  Google Scholar 

  3. 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/.

    Google Scholar 

  4. R. Beigel: When do extra majority gates help? polylog(-N) majority gates are equivalent to one, Comput. Complexity 4 (1994), 314–324.

    Article  MATH  MathSciNet  Google Scholar 

  5. 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.

    Article  MATH  MathSciNet  Google Scholar 

  6. 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.

    Google Scholar 

  7. R. Boppana and M. Sipser: The complexity of finite functions, in: Handbook of theoretical computer science, Vol. A, 757–804. Elsevier, Amsterdam, 1990.

    Google Scholar 

  8. A. Bogdanov and E. Viola: Pseudorandom bits for polynomials, SIAM J. on Computing 39 (2010), 2464–2486.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Baker and G. Wüstholz: Logarithmic forms and Diophantine geometry, volume 9 of New Mathematical Monographs, Cambridge University Press, 2007.

    Google Scholar 

  10. M. Braverman and O. Weinstein: A discrepancy lower bound for information complexity, in: Workshop on Randomization and Computation (RANDOM), 459–470, 2012.

    Google Scholar 

  11. 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.

    Google Scholar 

  12. I. Csiszár and J. Körner: Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, Inc., 1982.

    Google Scholar 

  13. A. Cobham: The intrinsic computational diffculty of functions, in: Int. Congress for Logic, Methodology and Philosophy of Science, 24–30, 1964.

    Google Scholar 

  14. F. R. K. Chung and P. Tetali: Communication complexity and quasi randomness, SIAM J. Discrete Math. 6 (1993), 110–123.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Dietzfelbinger, T. Hagerup, J. Katajainen and M. Penttonen: A reliable randomized algorithm for the closest-pair problem, J. Algorithms 25 (1997), 19–51.

    Article  MATH  MathSciNet  Google Scholar 

  16. 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.

    Google Scholar 

  17. U. Feige, P. Raghavan, D. Peleg and E. Upfal: Computing with noisy information, SIAM J. Comput. 23 (1994), 1001–1018.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. L. Furst, J. B. Saxe and M. Sipser: Parity, circuits, and the polynomial-time hierarchy, Mathematical Systems Theory 17 (1984), 13–27.

    Article  MATH  MathSciNet  Google Scholar 

  19. 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.

    Article  MATH  Google Scholar 

  20. O. Goldreich, S. Goldwasser and S. Micali: How to construct random functions, J. of the ACM 33 (1986), 792–807.

    Article  MathSciNet  Google Scholar 

  21. M. Goldmann, J. Håstad and A. A. Razborov: Majority gates vs. general weighted threshold gates, Computational Complexity 2 (1992), 277–300.

    Article  MATH  MathSciNet  Google Scholar 

  22. 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.

    Google Scholar 

  23. J. Håstad: Computational limitations of small-depth circuits, MIT Press, 1987.

    Google Scholar 

  24. J. Håstad: On the size of weights for threshold gates, SIAM J. Discrete Math. 7 (1994), 484–492.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Håstad and M. Goldmann: On the power of small-depth threshold circuits, Comput. Complexity 1 (1991), 113–129.

    Article  MATH  MathSciNet  Google Scholar 

  26. 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.

    Google Scholar 

  27. E. Kushilevitz and N. Nisan: Communication complexity, Cambridge University Press, 1997.

    Book  MATH  Google Scholar 

  28. L. A. Levin: One way functions and pseudorandom generators, Combinatorica 7 (1987), 357–363.

    Article  MATH  MathSciNet  Google Scholar 

  29. N. Linial, Y. Mansour and N. Nisan: Constant depth circuits, Fourier transform, and learnability, J. of the ACM 40 (1993), 607–620.

    MATH  MathSciNet  Google Scholar 

  30. N. Linial and A. Shraibman: Learning complexity vs communication complexity, Combinatorics, Probability & Computing 18 (2009), 227–245.

    Article  MATH  MathSciNet  Google Scholar 

  31. 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.

    Article  MATH  Google Scholar 

  32. S. Muroga, I. Toda and S. Takasu: Theory of majority decision elements, J. Franklin Inst. 271 (1961), 376–418.

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Muroga: Threshold logic and its applications, Wiley-Interscience, New York, 1971.

    MATH  Google Scholar 

  34. V. A. Nepomnjaščiĭ: Rudimentary predicates and Turing calculations, Soviet Mathematics-Doklady 11 (1970), 1462–1465.

    Google Scholar 

  35. N. Nisan: The communication complexity of threshold gates, in: Combinatorics, Paul Erdős is Eighty, number 1 in Bolyai Society Mathematical Studies, 301–315, 1993.

    Google Scholar 

  36. J. Naor and M. Naor: Small-bias probability spaces: effcient constructions and applications, SIAM J. on Computing 22 (1993), 838–856.

    Article  MATH  MathSciNet  Google Scholar 

  37. 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.

    Article  MATH  MathSciNet  Google Scholar 

  38. M. Naor and O. Reingold: Number-theoretic constructions of efficient pseudorandom functions, J. of the ACM 51 (2004), 231–262.

    Article  MATH  MathSciNet  Google Scholar 

  39. M. Naor, O. Reingold and A. Rosen: Pseudorandom functions and factoring, SIAM J. Comput. 31 (2002), 1383–1404.

    Article  MATH  MathSciNet  Google Scholar 

  40. N. Nisan and D. Zuckerman: Randomness is linear in space, J. of Computer and System Sciences 52 (1996), 43–52.

    Article  MATH  MathSciNet  Google Scholar 

  41. V. Podol’skiĭ: Perceptrons of large weight, Problems of Information Transmission 45 (2009), 46–53.

    Article  MathSciNet  Google Scholar 

  42. R. Raz: The BNS-Chung criterion for multi-party communication complexity, Comput. Complexity 9 (2000), 113–122.

    Article  MATH  MathSciNet  Google Scholar 

  43. A. Razborov and S. Rudich: Natural proofs, J. of Computer and System Sciences 55 (1997), 24–35.

    Article  MATH  MathSciNet  Google Scholar 

  44. D. V. Smirnov: Shannon’s information methods for lower bounds for probabilistic communication complexity, Master’s thesis, Moscow University, 1988.

    Google Scholar 

  45. H. Straubing and D. Théerien: A note on modp — modm circuits, Theory Comput. Syst. 39 (2006), 699–706.

    Article  MATH  MathSciNet  Google Scholar 

  46. R. Shaltiel and E. Viola: Hardness amplification proofs require majority, SIAM J. on Computing 39 (2010), 3122–3154.

    Article  MATH  MathSciNet  Google Scholar 

  47. E. Viola: Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates, SIAM J. on Computing 36 (2007), 1387–1403.

    Article  MATH  MathSciNet  Google Scholar 

  48. E. Viola: Cell-probe lower bounds for prefix sums, 2009, arXiv:0906.1370v1.

    Google Scholar 

  49. 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.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Northeastern University, Boston, MA, 02115, USA

    Emanuele Viola

Authors
  1. Emanuele Viola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emanuele Viola.

Additional information

Supported by NSF grants CCF-0845003, CCF-1319206.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Viola, E. The communication complexity of addition. Combinatorica 35, 703–747 (2015). https://doi.org/10.1007/s00493-014-3078-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-014-3078-3

Mathematics Subject Classification (2000)

Search

Navigation