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Efficient computation of middle levels Gray codes
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Mutze, Torsten and Nummenpalo, Jerri (2018) Efficient computation of middle levels Gray codes. ACM Transactions on Algorithms, 14 (2). 15. doi:10.1145/3170443
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Official URL: https://doi.org/10.1145/3170443
Abstract
For any integer $n\geq 1$ a \emph{middle levels Gray code} is a cyclic listing of all bitstrings of length $2n+1$ that have either $n$ or $n+1$ entries equal to 1 such that any two consecutive bitstrings in the list differ in exactly one bit.
The question whether such a Gray code exists for every $n\geq 1$ 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 this work we provide the first efficient algorithm to compute a middle levels Gray code.
For a given bitstring, our algorithm computes the next $\ell$ bitstrings in the Gray code in time $\cO(n\ell(1+\frac{n}{\ell}))$, which is $\cO(n)$ on average per bitstring provided that $\ell=\Omega(n)$.
| 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 | ||||||
| Library of Congress Subject Headings (LCSH): | Combinatorial analysis, Gray codes | ||||||
| Journal or Publication Title: | ACM Transactions on Algorithms | ||||||
| Publisher: | ACM | ||||||
| Official Date: | June 2018 | ||||||
| Dates: |
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| Volume: | 14 | ||||||
| Number: | 2 | ||||||
| Article Number: | 15 | ||||||
| DOI: | 10.1145/3170443 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Publisher Statement: | In connection with any use by the Owner of the Submitted Version (if accepted) or the Accepted Version or a Minor Revision, Owner shall use best efforts to display the ACM citation, along with a statement substantially similar to the following: © Author | ACM 2018. This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in ACM Transactions on Algorithms,, http://dx.doi.org/10.1145/3170443 | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Open Access Version: |
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