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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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WRAP-constant-time-algorithm-middle-levels-Gray-codes-Mutze-2020.pdf - Accepted Version - Requires a PDF viewer. Download (1276Kb) | Preview |
Official URL: https://doi.org/10.1007/s00453-019-00640-2
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 | ||||||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |
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| Divisions: | Faculty of Science > Computer Science | ||||||||||
| Journal or Publication Title: | Algorithmica | ||||||||||
| Publisher: | Springer Verlag | ||||||||||
| ISSN: | 0178-4617 | ||||||||||
| Official Date: | May 2020 | ||||||||||
| Dates: |
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| 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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