License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2017.49
URN: urn:nbn:de:0030-drops-69832
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/6983/
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Kulikov, Alexander S. ; Podolskii, Vladimir V.

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

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LIPIcs-STACS-2017-49.pdf (0.5 MB)

Abstract

We study the following computational problem: for which values of k, the majority of n bits MAJ_n 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 MAJ_k o MAJ_k. We observe that the minimum value of k for which there exists a MAJ_k o MAJ_k circuit that has high correlation with the majority of n bits is equal to Theta(sqrt(n)). We then show that for a randomized MAJ_k o MAJ_k circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n^(2/3+o(1)). We show a worst case lower bound: if a MAJ_k o MAJ_k circuit computes the majority of n bits correctly on all inputs, then k <= n^(13/19+o(1)). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k= O(n^(2/3)) can compute MAJ_n correctly on all inputs.

BibTeX - Entry

@InProceedings{kulikov_et_al:LIPIcs:2017:6983,
  author =	{Alexander S. Kulikov and Vladimir V. Podolskii},
  title =	{{Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{49:1--49:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Heribert Vollmer and Brigitte ValleĢe},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/6983},
  URN =		{urn:nbn:de:0030-drops-69832},
  doi =		{10.4230/LIPIcs.STACS.2017.49},
  annote =	{Keywords: circuit complexity, computational complexity, threshold, majority, lower bound, upper bound}
}

Keywords: circuit complexity, computational complexity, threshold, majority, lower bound, upper bound
Collection: 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)
Issue Date: 2017
Date of publication: 06.03.2017

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