computational complexity

, Volume 24, Issue 3, pp 645–694 | Cite as

The NOF Multiparty Communication Complexity of Composed Functions

  • Anil Ada
  • Arkadev ChattopadhyayEmail author
  • Omar Fawzi
  • Phuong Nguyen


We study the k-party “number on the forehead” communication complexity of composed functions \({f \circ \vec{g}}\), where \({f:\{0,1\}^n \to \{\pm 1\}}\), \({\vec{g} = (g_1,\ldots,g_n)}\), \({g_i : \{0,1\}^k \to \{0,1\}}\) and for \({(x_1,\ldots,x_k) \in (\{0,1\}^n)^k}\), \({f \circ \vec{g}(x_1,\ldots,x_k) = f(\ldots,g_i(x_{1,i},\ldots,x_{k,i}), \ldots)}\). When \({\vec{g} = (g, g,\ldots, g)}\), we denote \({f \circ \vec{g}}\) by \({f \circ g}\). We show that there is an \({O({\rm log}^3 n)}\) cost simultaneous protocol for SYM \({\circ g}\) when k >  1 + log  n, SYM is any symmetric function and g is any function. When k >  1 +  2 log  n, our simultaneous protocol applies to SYM \({\circ \, \vec{g}}\) with \({\vec{g}}\) being a vector of n arbitrary functions. We also get a non-simultaneous protocol for SYM \({\circ \, \vec{g}}\) of cost \({O(n/2^k \cdot {\rm log}\, n+ k {\rm log}\, n)}\) for any k ≥  2. In the setting of k ≤  1 + log  n, we study more closely functions of the form MAJORITY \({\circ g}\), MOD m \({\circ g}\) and NOR \({\circ g}\), where the latter two are generalizations of the well-known and studied functions generalized inner product and disjointness, respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137–166, 2003) and determine the communication complexity of MAJORITYQCSB k , where QCSB k is the “quadratic character of the sum of the bits” function.

In the second part of our paper, we utilize the connection between the ‘number on the forehead’ model and Ramsey theory to construct a large set without a k-dimensional corner (k-dimensional generalization of a k-term arithmetic progression) in \({(\mathbb{F}_{2}^{n})^k}\), thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [N] ×  [N] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost \({O(\sqrt{n})}\) for the EXACT N function. For x 1,x 2,x 3 n-bit integers, EXACT N (x 1,x 2,x 3) = −1 iff x 1x 2x 3 = N.


Number on the forehead model communication complexity Ramsey theory 

Subject Classification

68Q05 68Q17 05D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Noga Alon, Yossi Matias, Mario Szegedy (1999) The Space Complexity of Approximating the Frequency Moments. Journal of Computer and System Sciences 58: 137–147zbMATHMathSciNetCrossRefGoogle Scholar
  2. László Babai, Anna Gál, Peter G. Kimmel & Satyanarayana V. Lokam (2003). Communication Complexity of Simultaneous Messages. SIAM Journal on Computing 33, 137–166.Google Scholar
  3. László Babai, Peter G. Kimmel & Satyanarayana V. Lokam (1995). Simultaneous messages vs. communication. In In 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS), 361–372. Springer.Google Scholar
  4. László Babai, Noam Nisan, Mario Szegedy (1992) Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. Journal of Computer and System Sciences 45(2): 204–232zbMATHMathSciNetCrossRefGoogle Scholar
  5. Michael Bateman & Nets H. Katz (2011). New bounds on cap sets. URL
  6. Paul Beame, Matei David, Toniann Pitassi & Philipp Woelfel (2010). Separating Deterministic from Randomized Multiparty Communication Complexity. Theory of Computing 6(1), 201–225. URL
  7. Paul Beame& Dang-Trinh Huynh-Ngoc (2009). Multiparty Communication Complexity and Threshold Circuit Size of AC 0. In Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’09, 53–62. IEEE Computer Society, Washington, DC, USA. URL
  8. Felix A. Behrend (1946) On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. Proceedings of the National Academy of Sciences 32: 331–332MathSciNetCrossRefGoogle Scholar
  9. Richard Beigel, William Gasarch & James Glenn (2006). The Multiparty Communication Complexity of Exact T: Improved Bounds and New Problems. In Mathematical Foundations of Computer Science 2006, Rastislav Královic & Pawel Urzyczyn, editors, volume 4162 of Lecture Notes in Computer Science, 146–156. Springer Berlin Heidelberg. URL
  10. Richard Beigel, Jun Tarui (1994) On ACC. Computational Complexity 4: 350–366zbMATHMathSciNetCrossRefGoogle Scholar
  11. Eric Blais, Joshua Brody & Kevin Matulef (2011). Property Testing Lower Bounds via Communication Complexity. In In Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity (CCC).Google Scholar
  12. Jean Bourgain (1999). On triples in arithmetic progression. Geometric and Functional Analysis.Google Scholar
  13. Ashok K. Chandra, Merrick L. Furst & Richard J. Lipton (1983). Multi-party protocols. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, STOC ’83, 94–99. ACM, New York, NY, USA. URL
  14. Arkadev Chattopadhyay (2008). Circuits, Communication and Polynomials. Ph.D. thesis, McGill University.Google Scholar
  15. Arkadev Chattopadhyay & Anil Ada (2008). Multiparty communication complexity of disjointness. Technical report, In Electronic Colloquium on Computational Complexity (ECCC) TR08–002.Google Scholar
  16. Arkadev Chattopadhyay, Andreas Krebs, Michal Koucky, Mario Szegedy, Pascal Tesson & Denis Thérien (2007). Languages with bounded multiparty communication complexity. In Proceedings of the 24th annual conference on Theoretical aspects of computer science, STACS’07, 500–511. Springer-Verlag, Berlin, Heidelberg. URL
  17. Benny Chor & Oded Goldreich (1988). Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17(2), 230–261. ISSN 0097-5397. URL
  18. Fan R.K. Chung, Prasad Tetali (1993) Communication complexity and quasi randomness. SIAM Journal on Discrete Mathematics 6(1): 110–123zbMATHMathSciNetCrossRefGoogle Scholar
  19. Vincent Conitzer & Tuomas Sandholm (2004). Communication complexity as a lower bound for learning in games. In International Conference on Machine Learning.Google Scholar
  20. Michael Elkin (2011) An improved construction of progression-free sets. Israel Journal of Mathematics 184((1): 93–128zbMATHMathSciNetGoogle Scholar
  21. Jürgen Forster, Matthias Krause, Satyanarayana V. Lokam, Rustam Mubarakzjanov, Niels Schmitt & Hans ulrich Simon (2001). Relations Between Communication Complexity, Linear Arrangements, and Computational Complexity. In Foundations of Software Technology and Theoretical Computer Science, 171–182.Google Scholar
  22. Harry Furstenberg, Yitzhak Katznelson (1978) An ergodic Szemerédi theorem for commuting transformations. Journal d’Analyse Mathematique 34: 275–291MathSciNetCrossRefGoogle Scholar
  23. W. Timothy Gowers (2001). A new proof of Szemerédi’s theorem. Geometric and Functional Analysis 11, 465–588.Google Scholar
  24. W. Timothy Gowers (2007). Hypergraph regularity and the multidimensional Szemerédi theorem. Annals of Mathematics 166, 897–946.Google Scholar
  25. W. Timothy Gowers (2010). Decompositions, approximate structure, transference, and the Hahn-Banach theorem. Bulletin of the London Mathematical Society 42, 573–606.Google Scholar
  26. Ben Green (2005). Surveys in Combinatorics 2005, chapter Finite field models in additive combinatorics, 1–27. London Math. Soc. Lecture Notes 327. Cambridge Univ Press.Google Scholar
  27. Vince Grolmusz (1994) The BNSLower Bound for Multi-party Protocols Is Nearly Optimal. Information and Computation 112: 51–54zbMATHMathSciNetCrossRefGoogle Scholar
  28. Vince Grolmusz (1995). Separating the Communication Complexities of MOD m and MOD p Circuits. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science (FOCS), 278–287.Google Scholar
  29. Vince Grolmusz (1998) Circuits and Multi-Party Protocols. Computational Complexity 7: 1–18zbMATHMathSciNetCrossRefGoogle Scholar
  30. Johan Håstad, Mikael Goldmann (1991) On The Power Of Small-Depth Threshold Circuits. Computational Complexity 1: 610–618Google Scholar
  31. Hartmut Klauck (2007) Lower Bounds for Quantum Communication Complexity. Siam Journal on Computing 37: 20–46zbMATHMathSciNetCrossRefGoogle Scholar
  32. Eyal Kushilevitz & Noam Nisan (1997). Communication complexity. Cambridge University Press.Google Scholar
  33. Michael T. Lacey & William McClain (2007). On an Argument of Shkredov on Two-Dimensional Corners. Online Journal of Analytic Combinatorics.Google Scholar
  34. Troy Lee, Gideon Schechtman & Adi Shraibman (2009). Lower bounds on quantum multiparty communication complexity. In In Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC), 254–262.Google Scholar
  35. Troy Lee, Adi Shraibman (2009) Disjointness is Hard in the Multiparty Number-on-the-Forehead Model. Computational Complexity 18: 309–336zbMATHMathSciNetCrossRefGoogle Scholar
  36. Peter Bro Miltersen, Noam Nisan, Shmuel Safra & Avi Wigderson (1998). On Data Structures and Asymmetric Communication Complexity. Journal of Computer and System Sciences 57(1), 37–49. URL Scholar
  37. Ashley Montanaro & Tobias Osborne (2009). On the communication complexity of XOR functions. Arxiv preprint arXiv:0909.3392.Google Scholar
  38. N. Nisan (1993). The communication complexity of threshold gates. Combinatorica.Google Scholar
  39. Noam Nisan, Ilya Segal (2006) The communication requirements of efficient allocations and supporting prices. Journal of Economic Theory 129: 192–224zbMATHMathSciNetCrossRefGoogle Scholar
  40. Noam Nisan, Avi Wigderson (1993) Rounds in Communication Complexity Revisited. SIAM Journal on Computing 22: 211–219zbMATHMathSciNetCrossRefGoogle Scholar
  41. Kevin O’Bryant (2011). Sets of integers that do not contain long arithmetic progressions. The Electronic Journal of Combinatorics 18(1), 59.Google Scholar
  42. Beame Paul, Toniann Pitassi & Nathan Segerlind (2007). Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity. SIAM Journal on Computing 37, 845–869. URL Scholar
  43. Pavel Pudlák (2003). An Application of Hindman’s Theorem to a Problem on Communication Complexity. Combinatorics, Probability & Computing 12, 661–670. URL
  44. Pavel Pudlák (2006). Personal communication.Google Scholar
  45. Ran Raz (1995). Fourier Analysis for Probabilistic Communication Complexity. Computational Complexity 5, 205–221.Google Scholar
  46. Ran Raz (2000) The BNS-Chung criterion for multi-party communication complexity. Computational Complexity 9(2): 113–122zbMATHMathSciNetCrossRefGoogle Scholar
  47. Alexander Razborov (2003) Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics 67(1): 145–159MathSciNetCrossRefGoogle Scholar
  48. Klaus F. Roth (1953). On Certain Sets of Integers. Journal of The London Mathematical Society-second Series s1-28, 104–109.Google Scholar
  49. Tom Sanders (2011) On Roths theorem on progressions. Annals of Mathematics 174: 619–636zbMATHMathSciNetCrossRefGoogle Scholar
  50. Alexander A. Sherstov (2007). The pattern matrix method for lower bounds on quantum communication. In In Proceedings of the 40th Symposium on Theory of Computing (STOC), 85–94.Google Scholar
  51. Alexander A. Sherstov (2012). The multiparty communication complexity of set disjointness. In Proceedings of the 44th symposium on Theory of Computing, STOC ’12, 525–548. ACM, New York, NY, USA. URL
  52. Yaoyun Shi, Zhiqiang Zhang (2009) Communication complexities of symmetric XOR functions. Quantum Information and Computation 9: 255–263zbMATHMathSciNetGoogle Scholar
  53. Yaoyun Shi & Yufan Zhu (2009). Quantum communication complexity of block-composed functions. Quantum Information and Computation 9, 444–460. URL
  54. Ilya D. Shkredov (2006a). On a Generalization of Szemerédi’s Theorem. Proc. London Math. Soc. 93, 723–760.Google Scholar
  55. Ilya D. Shkredov (2006b). On a problem of Gowers. Izvestiya: Mathematics 70, 385.Google Scholar
  56. Victor Shoup (2009). A computational introduction to number theory and algebra. Cambridge University Press.Google Scholar
  57. Roman Smolensky (1987). Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, STOC ’87, 77–82. ACM, New York, NY, USA. URL
  58. Pascal Tesson (2003). Computational complexity questions related to finite semigroups and monoids. Ph.D. thesis, McGill University.Google Scholar
  59. Emanuele Viola & Avi Wigderson (2008). Norms, XOR Lemmas, and Lower Bounds for Polynomials and Protocols. Theory of Computing 4(1), 137–168. URL
  60. Andrew C. Yao (1979). Some complexity questions related to distributive computing (Preliminary Report). In Proceedings of the eleventh annual ACM symposium on Theory of computing, 209–213. ACM Press, New York, NY, USA.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Anil Ada
    • 1
  • Arkadev Chattopadhyay
    • 2
    Email author
  • Omar Fawzi
    • 3
  • Phuong Nguyen
    • 4
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.School of Technology and Computer ScienceTata Institute of Fundamental ResearchMumbaiIndia
  3. 3.Institute for Theoretical Physics, ETHZürichSwitzerland
  4. 4.Dép. d’informatique et de recherche opérationnelleUniversité de MontréalMontrealCanada

Personalised recommendations