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.ICALP.2018.66
URN: urn:nbn:de:0030-drops-90703
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9070/
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Gregor, Petr ; Jäger, Sven ; Mütze, Torsten ; Sawada, Joe ; Wille, Kaja

Gray Codes and Symmetric Chains

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LIPIcs-ICALP-2018-66.pdf (0.6 MB)


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.

BibTeX - Entry

@InProceedings{gregor_et_al:LIPIcs:2018:9070,
  author =	{Petr Gregor and Sven J{\"a}ger and Torsten M{\"u}tze and Joe Sawada and Kaja Wille},
  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 =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9070},
  URN =		{urn:nbn:de:0030-drops-90703},
  doi =		{10.4230/LIPIcs.ICALP.2018.66},
  annote =	{Keywords: Gray code, Hamilton cycle, hypercube, poset, symmetric chain}
}

Keywords: Gray code, Hamilton cycle, hypercube, poset, symmetric chain
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018


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