On universal partial words
A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systemati...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2017-05-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2205/pdf |
| _version_ | 1827323758341259264 |
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| author | Herman Z. Q. Chen Sergey Kitaev Torsten Mütze Brian Y. Sun |
| author_facet | Herman Z. Q. Chen Sergey Kitaev Torsten Mütze Brian Y. Sun |
| author_sort | Herman Z. Q. Chen |
| collection | DOAJ |
| description | A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a
word over $A$ such that every word in $A^n$ appears exactly once as a subword
(cyclically or linearly). It is well-known and easy to prove that universal
words exist for any $A$ and $n$. In this work we initiate the systematic study
of universal partial words. These are words that in addition to the letters
from $A$ may contain an arbitrary number of occurrences of a special `joker'
symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For
example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet
$A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the
subwords $000$ and $010$). We present results on the existence and
non-existence of linear and cyclic universal partial words in different
situations (depending on the number of $\Diamond$s and their positions),
including various explicit constructions. We also provide numerous examples of
universal partial words that we found with the help of a computer. |
| first_indexed | 2024-04-25T01:58:45Z |
| format | Article |
| id | doaj.art-9f352357fbb2482b98858b17c4afd2e0 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T01:58:45Z |
| publishDate | 2017-05-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-9f352357fbb2482b98858b17c4afd2e02024-03-07T15:32:48ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502017-05-01Vol. 19 no. 1Combinatorics10.23638/DMTCS-19-1-162205On universal partial wordsHerman Z. Q. ChenSergey KitaevTorsten MützeBrian Y. SunA universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer.https://dmtcs.episciences.org/2205/pdfmathematics - combinatoricscomputer science - formal languages and automata theorycomputer science - information theory |
| spellingShingle | Herman Z. Q. Chen Sergey Kitaev Torsten Mütze Brian Y. Sun On universal partial words Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics computer science - formal languages and automata theory computer science - information theory |
| title | On universal partial words |
| title_full | On universal partial words |
| title_fullStr | On universal partial words |
| title_full_unstemmed | On universal partial words |
| title_short | On universal partial words |
| title_sort | on universal partial words |
| topic | mathematics - combinatorics computer science - formal languages and automata theory computer science - information theory |
| url | https://dmtcs.episciences.org/2205/pdf |
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