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DOI: 10.4230/LIPIcs.ICALP.2018.66

Gray Codes and Symmetric Chains

Authors: Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, and Kaja Wille

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract

We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-l,n+l], where 1 <= l <= n+1, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (l=1). We provide a solution for the case l=2 and solve a relaxed version of the problem for general values of l, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the n-dimensional hypercube for any n >= 12.

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Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, and Kaja Wille. Gray Codes and Symmetric Chains. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gregor_et_al:LIPIcs.ICALP.2018.66,
  author =	{Gregor, Petr and J\"{a}ger, Sven and M\"{u}tze, Torsten and Sawada, Joe and Wille, Kaja},
  title =	{{Gray Codes and Symmetric Chains}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{66:1--66:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.66},
  URN =		{urn:nbn:de:0030-drops-90703},
  doi =		{10.4230/LIPIcs.ICALP.2018.66},
  annote =	{Keywords: Gray code, Hamilton cycle, hypercube, poset, symmetric chain}
}

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Gray Codes and Symmetric Chains

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