On recognising words that are squares for the shuffle product

Romeo Rizzi; Stéphane Vialette

doi:10.1016/j.tcs.2017.04.003

Article Dans Une Revue Theoretical Computer Science Année : 2023

On recognising words that are squares for the shuffle product

(1) , (2)

1
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Romeo Rizzi

Fonction : Auteur
PersonId : 1011885

Università degli studi di Verona = University of Verona

Stéphane Vialette

Fonction : Auteur
PersonId : 3062
IdHAL : stephane-vialette
ORCID : 0000-0003-2308-6970
IdRef : 061620734

Laboratoire d'Informatique Gaspard-Monge

Résumé

The shuffle of two words u and v of

A^{⁎}

is the language

u ⧢ v

consisting of all words

u_{1} v_{1} u_{2} v_{2} \dots u_{k} v_{k}

, where

k \geq 0

and the

u_{i}

and

v_{i}

are words of

A^{⁎}

such that

u = u_{1} u_{2} \dots u_{k}

and

v = v_{1} v_{2} \dots v_{k}

. In other words,

u ⧢ v

is the finite set of all words obtainable from merging the words u and v from left to right, but choosing the next symbol arbitrarily from u or v. A word

u \in A^{⁎}

is a square for the shuffle product if it is the shuffle of two identical words (i.e.,

u \in v ⧢ v

for some

v \in A^{⁎}

Whereas it can be decided in polynomial-time whether or not $u \in v_{1} ⧢ v_{2}$ for given words u, $v_{1}$ and $v_{2}$ (J.-C. Spehner, 1986), we show in this paper that it is NP-complete to determine whether or not a word u is a square for the shuffle product. The novelty in our approach lies in representing words as linear graphs, in which deciding whether or not a given word is a square for the shuffle product reduces to computing some inclusion-free perfect matching. Finally, we prove that it is NP-complete to determine whether or not an input word is in the shuffle of a word with its reverse.

Mots clés

combinatorics on words shuffle operator complexity

Domaines

Algorithme et structure de données [cs.DS]

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RizziVialetteTCS2023.pdf (413.44 Ko)

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https://hal.science/hal-04498180

Soumis le : lundi 11 mars 2024-11:19:23

Dernière modification le : mercredi 30 octobre 2024-18:18:01

Archivage à long terme le : mercredi 12 juin 2024-18:39:08

Dates et versions

hal-04498180 , version 1 (11-03-2024)

Identifiants

HAL Id : hal-04498180 , version 1
DOI : 10.1016/j.tcs.2017.04.003

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Romeo Rizzi, Stéphane Vialette. On recognising words that are squares for the shuffle product. Theoretical Computer Science, 2023, 956, pp.111156.1-16. ⟨10.1016/j.tcs.2017.04.003⟩. ⟨hal-04498180⟩

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On recognising words that are squares for the shuffle product

Résumé

Mots clés

Domaines

Dates et versions

Identifiants

Citer

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