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On a combinatorial generation problem of Knuth
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Merino, Arturo, Micka, Ondrej and Mutze, Torsten (2021) On a combinatorial generation problem of Knuth. SIAM Journal on Computing . (In Press)
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Abstract
The well-known middle levels conjecture asserts that for every integer n ≥ 1, all binary strings of length 2(n + 1) with exactly n + 1 many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book ‘The Art of Computer Programming Vol. 4A’ Knuth raised a stronger form of this conjecture (Problem 56 in Section 7.2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has 2n + 1 blocks of the same length, and each block is obtained by adding s = 1 (modulo 2n + 1) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts s ≥ 1 that are coprime to 2n + 1. We also present an algorithm to compute this ordering, generating each new bitstring in (n) time, using (n) memory in total.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Combinatorial analysis, Computational complexity, Computer science -- Mathematics | ||||||||||||
| Journal or Publication Title: | SIAM Journal on Computing | ||||||||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||||||||
| ISSN: | 0097-5397 | ||||||||||||
| Official Date: | 2021 | ||||||||||||
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| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | In Press | ||||||||||||
| Reuse Statement (publisher, data, author rights): | “First Published in SIAM Journal on Computing in [volume and number, or year], published by the Society for Industrial and Applied Mathematics (SIAM)” | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
| Copyright Holders: | © by SIAM. Unauthorized reproduction of this article is prohibited | ||||||||||||
| Description: | An extended abstract of this paper has been accepted for presentation at the 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2021) |
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| Version or Related Resource: | https://epubs.siam.org/doi/epdf/10.1137/1.9781611976465.46 | ||||||||||||
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