Theory of Computing Systems

, Volume 63, Issue 5, pp 956–986 | Cite as

Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates

  • Alexander S. KulikovEmail author
  • Vladimir V. Podolskii
Part of the following topical collections:
  1. Special Issue on Theoretical Aspects of Computer Science (STACS 2017)


We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk ∘ MAJk. We observe that the minimum value of k for which there exists a MAJk ∘ MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk ∘ MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3 + o(1). We show a worst case lower bound: if a MAJk ∘ MAJk circuit computes the majority of n bits correctly on all inputs, then kn13/19 + o(1).


Circuit complexity Constant depth Majority Threshold Correlation Average case Worst case 



We would like to thank the participants of Low-Depth Complexity Workshop (St. Petersburg, Russia, May 21–25, 2016) for many helpful discussions.


  1. 1.
    Allender, E., Koucký, M.: Amplifying lower bounds by means of self-reducibility. J. ACM 57(3), 14:1–14:36 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amano, K., Yoshida, M.: Depth two (n − 2)-majority circuits for n-majority. Preprint (2017)Google Scholar
  3. 3.
    Bruno, B.: Personal communication (2017)Google Scholar
  4. 4.
    Xi, C., Oliveira, I.C., Servedio, R.A.: Addition is exponentially harder than counting for shallow monotone circuits. Electronic Colloquium on Computational Complexity (ECCC) 22, 123 (2015)Google Scholar
  5. 5.
    Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  6. 6.
    Engels, C., Garg, M., Makino, K., Rao, A.: On expressing majority as a majority of majorities. Electronic Colloquium on Computational Complexity (ECCC) 24, 174 (2017)Google Scholar
  7. 7.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968)Google Scholar
  8. 8.
    Goldmann, M., Håstad, J., Razborov, A.A.: Majority gates vs. general weighted threshold gates. Comput. Complex. 2, 277–300 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Goldreich, O.: Valiant’s polynomial-size monotone formula for majority, 2001. Available at
  10. 10.
    Hofmeister, T.: The power of negative thinking in constructing threshold circuits for addition. In: Proceedings of the Seventh Annual Structure in Complexity Theory Conference, Boston, Massachusetts, USA, June 22-25, 1992, pp 20–26 (1992)Google Scholar
  11. 11.
    Hush, D., Scovel, C.: Concentration of the hypergeometric distribution. Stat. Probab. Lett. 75(2), 127–132 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jukna, S.: Extremal Combinatorics - With Applications in Computer Science. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2011)Google Scholar
  13. 13.
    Jukna, S.: Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, Berlin (2012)Google Scholar
  14. 14.
    Jukna, S., Razborov, A.A., Savický, P., Wegener, I.: On P versus NP cap co-NP for decision trees and read-once branching programs. Comput. Complex. 8(4), 357–370 (1999)CrossRefGoogle Scholar
  15. 15.
    Kamp, J., Zuckerman, D.: Deterministic extractors for bit-fixing sources and exposure-resilient cryptography. SIAM J. Comput. 36(5), 1231–1247 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kane, D.M., Williams, R.: Super-linear gate and super-quadratic wire lower bounds for depth-two and depth-three threshold circuits. In: Wichs, D., Mansour, Y. (eds.) Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pp 633-643 (2016)Google Scholar
  17. 17.
    Kombarov, Y.A.: On depth two circuits for the majority function. In: Proceedings of Problems in theoretical cybernetics, pp 129–132. Max Press (2017)Google Scholar
  18. 18.
    Kulikov, A.S., Podolskii, V.V.: Computing majority by constant depth majority circuits with low fan-in gates. In: Vollmer, H., Vallée, B. (eds.) 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11 Hannover, Germany, volume 66 of LIPIcs, p 2017. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  19. 19.
    Magniez, F., Nayak, A., Santha, M., Sherman, J., Tardos, G., Xiao, D.: Improved bounds for the randomized decision tree complexity of recursive majority. Random Struct. Algorithms 48(3), 612–638 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Minsky, M., Papert, S.: Perceptrons - An Introduction to Computational Geometry. MIT Press, Cambridge (1987)Google Scholar
  21. 21.
    Mossel, E., O’Donnell, R.: On the noise sensitivity of monotone functions. Random Struct. Algorithms 23(3), 333–350 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  23. 23.
    Posobin, G.: Computing majority with low-fan-in majority queries. CoRR, arXiv:1711.10176 (2017)
  24. 24.
    Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms. Addison-Wesley-Longman, Reading (1996)Google Scholar
  25. 25.
    Serfling, R.J.: Probability inequalities for the sum in sampling without replacement. Ann. Stat. 2(1), 39–48 (1974)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Siu, K.-Y., Bruck, J.: On the power of threshold circuits with small weights. SIAM J. Discrete Math. 4(3), 423–435 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Valiant, L.G.: Short monotone formulae for the majority function. J. Algorithms 5(3), 363–366 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute at St. PetersburgRussian Academy of SciencesSt. PetersburgRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of Sciences and National Research University Higher School of EconomicsMoscowRussia

Personalised recommendations