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.2020.60
URN: urn:nbn:de:0030-drops-124678
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12467/
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Gregor, Petr ; Mička, Ondřej ; Mütze, Torsten

On the Central Levels Problem

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LIPIcs-ICALP-2020-60.pdf (0.8 MB)

Abstract

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-? many 1s and at most m+? many 1s, i.e., the vertices in the middle 2? levels, has a Hamilton cycle for any m ≥ 1 and 1 ≤ ? ≤ m+1. This problem was raised independently by Savage, by Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case ? = 1, and classical binary Gray codes, namely the case ? = m+1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of ? consecutive levels in the n-dimensional hypercube for any n ≥ 1 and 1 ≤ ? ≤ n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, n≥ 2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.

BibTeX - Entry

@InProceedings{gregor_et_al:LIPIcs:2020:12467,
  author =	{Petr Gregor and Ondřej Mička and Torsten M{\"u}tze},
  title =	{{On the Central Levels Problem}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{60:1--60:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12467},
  URN =		{urn:nbn:de:0030-drops-124678},
  doi =		{10.4230/LIPIcs.ICALP.2020.60},
  annote =	{Keywords: Gray code, Hamilton cycle, hypercube, middle levels, symmetric chain decomposition}
}

Keywords: Gray code, Hamilton cycle, hypercube, middle levels, symmetric chain decomposition
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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