Star Transposition Gray Codes for Multiset Permutations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10450676" target="_blank" >RIV/00216208:11320/22:10450676 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4230/LIPIcs.STACS.2022.34" target="_blank" >https://doi.org/10.4230/LIPIcs.STACS.2022.34</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.STACS.2022.34" target="_blank" >10.4230/LIPIcs.STACS.2022.34</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Star Transposition Gray Codes for Multiset Permutations

Popis výsledku v původním jazyce

Given integers k &gt;= 2 and a1, . . ., ak &gt;= 1, let a := (a1, . . ., ak) and n := a1 + . . . + ak. An a-multiset permutation is a string of length n that contains exactly ai symbols i for each i = 1, . . ., k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a1 = . . . = ak = 1) can be generated by star transpositions, while combinations (k = 2) can be generated by these operations if and only if they are balanced (a1 = a2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Delta(a):= n - 2 max{a1, . . ., ak} that allows us to distinguish three different regimes for this problem. We show that if Delta(a) &lt; 0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Delta(a) &gt; 0, assuming that they exist for the case INCREMENT (a) = 0. For the case Delta(a) = 0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.

Název v anglickém jazyce

Star Transposition Gray Codes for Multiset Permutations

Popis výsledku anglicky

Given integers k &gt;= 2 and a1, . . ., ak &gt;= 1, let a := (a1, . . ., ak) and n := a1 + . . . + ak. An a-multiset permutation is a string of length n that contains exactly ai symbols i for each i = 1, . . ., k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a1 = . . . = ak = 1) can be generated by star transpositions, while combinations (k = 2) can be generated by these operations if and only if they are balanced (a1 = a2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Delta(a):= n - 2 max{a1, . . ., ak} that allows us to distinguish three different regimes for this problem. We show that if Delta(a) &lt; 0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Delta(a) &gt; 0, assuming that they exist for the case INCREMENT (a) = 0. For the case Delta(a) = 0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.

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)

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)

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Leibniz International Proceedings in Informatics, LIPIcs

ISBN

978-3-95977-222-8

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

14

Strana od-do

1-14

Název nakladatele

Schloss Dagstuhl - Leibniz-Zentrum fur Informatik

Místo vydání

Dagstuhl, Germany

Místo konání akce

Virtual, Marseille

Datum konání akce

15. 5. 2022

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

—