On a combinatorial generation problem of Knuth
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10450764" target="_blank" >RIV/00216208:11320/22:10450764 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=XwGrV0yTb-" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=XwGrV0yTb-</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/20M1377394" target="_blank" >10.1137/20M1377394</a>
Alternative languages
Result language
angličtina
Original language name
On a combinatorial generation problem of Knuth
Original language description
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 [Knu11] 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 O(n) time, using O(n) memory in total.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/GA19-08554S" target="_blank" >GA19-08554S: Structures and algorithms in highly symmetric graphs</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Computing
ISSN
0097-5397
e-ISSN
1095-7111
Volume of the periodical
51
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
45
Pages from-to
379-423
UT code for WoS article
000797465600002
EID of the result in the Scopus database
2-s2.0-85130586992