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A constant-time algorithm for middle levels Gray codes

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Mutze, Torsten and Nummenpalo, Jerri (2020) A constant-time algorithm for middle levels Gray codes. Algorithmica, 82 . pp. 1239-1258. doi:10.1007/s00453-019-00640-2

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Official URL: https://doi.org/10.1007/s00453-019-00640-2

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Abstract

For any integer $n\geq 1$ a \emph{middle levels Gray code} is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive subsets differ in adding or removing a single element.
The question whether such a Gray code exists for any $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T.~Mütze. Proof of the middle levels conjecture. \textit{Proc. London Math. Soc.}, 112(4):677--713, 2016].
In a follow-up paper [T.~Mütze and J.~Nummenpalo. An efficient algorithm for computing a middle levels Gray code. \textit{Proc. ESA}, 2015] this existence proof was turned into an algorithm that computes each new set in the Gray code in time $\mathcal{O}(n)$ on average.
In this work we complete this line of research by presenting an algorithm for computing a middle levels Gray code in optimal time and space: Each new set is generated in time $\mathcal{O}(1)$, and the required space is $\mathcal{O}(n)$.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science, Engineering and Medicine > Science > Computer Science
Journal or Publication Title: Algorithmica
Publisher: Springer Verlag
ISSN: 0178-4617
Official Date: May 2020
Dates:
DateEvent
May 2020Published
25 October 2019Available
10 October 2019Accepted
7 June 2018Submitted
Volume: 82
Page Range: pp. 1239-1258
DOI: 10.1007/s00453-019-00640-2
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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