On a combinatorial generation problem of Knuth
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10435488" target="_blank" >RIV/00216208:11320/21:10435488 - isvavai.cz</a>
Výsledek na webu
<a href="https://doi.org/10.1137/1.9781611976465.46" target="_blank" >https://doi.org/10.1137/1.9781611976465.46</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/1.9781611976465.46" target="_blank" >10.1137/1.9781611976465.46</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
On a combinatorial generation problem of Knuth
Popis výsledku v původním jazyce
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 O(n) time, using O(n) memory in total.
Název v anglickém jazyce
On a combinatorial generation problem of Knuth
Popis výsledku anglicky
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 O(n) time, using O(n) memory in total.
Klasifikace
Druh
D - Stať ve sborníku
CEP obor
—
OECD FORD obor
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Návaznosti výsledku
Projekt
<a href="/cs/project/GA19-08554S" target="_blank" >GA19-08554S: Struktury a algoritmy ve velmi symetrických grafech</a><br>
Návaznosti
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Ostatní
Rok uplatnění
2021
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název statě ve sborníku
Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms
ISBN
978-1-61197-646-5
ISSN
—
e-ISSN
—
Počet stran výsledku
9
Strana od-do
735-743
Název nakladatele
Society for Industrial and Applied Mathematics
Místo vydání
Philadelphia, USA
Místo konání akce
Alexandria, Virginia, USA (Virtual Conference)
Datum konání akce
10. 1. 2021
Typ akce podle státní příslušnosti
WRD - Celosvětová akce
Kód UT WoS článku
—