On shuffled-square-free words
A word u is a shuffle of words v and w, which we denote by u∈v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the left-to-right ordering of the letters from v and w. In case u∈v⧢v for some word v, the word u is called a shuffled-square. A word u is shuffled-square...
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| Published in: | Theoretical computer science Vol. 941; pp. 91 - 103 |
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04-01-2023
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| Abstract | A word u is a shuffle of words v and w, which we denote by u∈v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the left-to-right ordering of the letters from v and w. In case u∈v⧢v for some word v, the word u is called a shuffled-square. A word u is shuffled-square-free if it does not have a non-empty factor (i.e., non-empty sequence of adjacent letters) that is a shuffled-square. Our contribution in this context is two-fold. First, we prove that there exist arbitrarily long shuffled-square-free words in any alphabet with six letters or more, thereby improving on a previous result of Guégan and Ochem. Furthermore, we show that recognizing shuffled-square-free words on arbitrary alphabets is NP-complete. |
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| AbstractList | A word u is a shuffle of words v and w, which we denote by u∈v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the left-to-right ordering of the letters from v and w. In case u∈v⧢v for some word v, the word u is called a shuffled-square. A word u is shuffled-square-free if it does not have a non-empty factor (i.e., non-empty sequence of adjacent letters) that is a shuffled-square. Our contribution in this context is two-fold. First, we prove that there exist arbitrarily long shuffled-square-free words in any alphabet with six letters or more, thereby improving on a previous result of Guégan and Ochem. Furthermore, we show that recognizing shuffled-square-free words on arbitrary alphabets is NP-complete. A word u is a shuffle of words v and w, which we denote by u ∈ v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the left-to-right ordering of the letters from v and w. In case u∈v⧢v for some word v, the word u is called a shuffled-square. A word u is shuffled-square-free if it does not have a non-empty factor (i.e., non-empty sequence of adjacent letters) that is a shuffled-square. Our contribution in this context is two-fold. First, we prove that there exist arbitrarily long shuffled-square-free words in any alphabet with six letters or more, thereby improving on a previous result of Guégan and Ochem. Furthermore, we show that recognizing shuffled-square-free words on arbitrary alphabets is NP-complete. |
| Author | Jugé, Vincent Bulteau, Laurent Vialette, Stéphane |
| Author_xml | – sequence: 1 givenname: Laurent surname: Bulteau fullname: Bulteau, Laurent email: laurent.bulteau@univ-eiffel.fr – sequence: 2 givenname: Vincent surname: Jugé fullname: Jugé, Vincent email: vincent.juge@univ-eiffel.fr – sequence: 3 givenname: Stéphane orcidid: 0000-0003-2308-6970 surname: Vialette fullname: Vialette, Stéphane email: stephane.vialette@univ-eiffel.fr |
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Skr. contributor: fullname: Thue – ident: 10.1016/j.tcs.2022.10.028_br0240 – volume: 3 start-page: 121 issue: 2–3 year: 1927 ident: 10.1016/j.tcs.2022.10.028_br0160 article-title: Über eine Schlussweise aus dem Endlichen ins Unendliche publication-title: Acta Litt. Sci. Reg. Univ. Hung. Francisco-Josephinae, Sect. Sci. Math. contributor: fullname: Kőnig – year: 2015 ident: 10.1016/j.tcs.2022.10.028_br0200 contributor: fullname: Müller – ident: 10.1016/j.tcs.2022.10.028_br0100 – volume: 4 start-page: 28 issue: 3 year: 2014 ident: 10.1016/j.tcs.2022.10.028_br0090 article-title: Combinatorics and algorithmics of strings (Dagstuhl seminar 14111) publication-title: Dagstuhl Rep. contributor: fullname: Crochemore – start-page: 329 year: 1997 ident: 10.1016/j.tcs.2022.10.028_br0070 article-title: Combinatorics of words contributor: fullname: Choffrut – year: 2005 ident: 10.1016/j.tcs.2022.10.028_br0170 contributor: fullname: Lothaire – volume: vol. 11 start-page: 65 year: 1992 ident: 10.1016/j.tcs.2022.10.028_br0020 article-title: Axel Thue's papers on repetitions in words: a translation contributor: fullname: Berstel – volume: 27 issue: 1 year: 2020 ident: 10.1016/j.tcs.2022.10.028_br0130 article-title: Extremal square-free words publication-title: Electron. J. Comb. contributor: fullname: Grytczuk – volume: 380 start-page: 373 issue: 3 year: 2007 ident: 10.1016/j.tcs.2022.10.028_br0150 article-title: A note on the number of squares in a word publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2007.03.025 contributor: fullname: Ilie |
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| Snippet | A word u is a shuffle of words v and w, which we denote by u∈v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the... A word u is a shuffle of words v and w, which we denote by u ∈ v⧢w, if u can be obtained by mixing the letters from v and w in a way that preserves the... |
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