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.MFCS.2018.81
URN: urn:nbn:de:0030-drops-96633
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9663/
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Amano, Kazuyuki

Depth Two Majority Circuits for Majority and List Expanders

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Abstract

Let MAJ_n denote the Boolean majority function of n input variables. In this paper, we study the construction of depth two circuits computing MAJ_n where each gate in a circuit computes MAJ_m for m < n.
We first give an explicit construction of depth two MAJ_{floor[n/2]+2} o MAJ_{<= n-2} circuits computing MAJ_n for every n >= 7 such that n congruent 3 (mod 4) where MAJ_m and MAJ_{<= m} denote the majority gates that take m and at most m distinct inputs, respectively. A graph theoretic argument developed by Kulikov and Podolskii (STACS '17, Article No. 49) shows that there is no MAJ_{<= n-2} o MAJ_{n-2} circuit computing MAJ_n. Hence, our construction reveals that the use of a smaller fan-in gates at the bottom level is essential for the existence of such a circuit. Some computational results are also provided.
We then show that the construction of depth two MAJ_m o MAJ_m circuits computing MAJ_n for m<n can be translated into the construction of a newly introduced version of bipartite expander graphs which we call a list expander. Intuitively, a list expander is a c-leftregular bipartite graph such that for a given d < c, every d-leftregular subgraph of the original graph has a certain expansion property. We formalize this connection and verify that, with high probability, a random bipartite graph is a list expander of certain parameters. However, the parameters obtained are not sufficient to give us a MAJ_{n-c} o MAJ_{n-c} circuit computing MAJ_n for a large constant c.

BibTeX - Entry

@InProceedings{amano:LIPIcs:2018:9663,
  author =	{Kazuyuki Amano},
  title =	{{Depth Two Majority Circuits for Majority and List Expanders}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{81:1--81:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9663},
  URN =		{urn:nbn:de:0030-drops-96633},
  doi =		{10.4230/LIPIcs.MFCS.2018.81},
  annote =	{Keywords: Boolean function, majority function, constant depth circuit, expander graph}
}

Keywords: Boolean function, majority function, constant depth circuit, expander graph
Collection: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue Date: 2018
Date of publication: 27.08.2018

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