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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
Article
Part of the following topical collections:
  1. Special Issue on Theoretical Aspects of Computer Science (STACS 2017)

Abstract

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).

Keywords

Circuit complexity Constant depth Majority Threshold Correlation Average case Worst case 

Notes

Acknowledgments

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

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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

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