### Abstract

Two strings x and y are said to be Abelian equivalent if x is a permutation of y, or vice versa. If a string z satisfies z = xy with x and y being Abelian equivalent, then z is said to be an Abelian square. If a string w can be factorized into a sequence v_{1}, …, vs of strings such that v_{1}, …, v_{s-1} are all Abelian equivalent and vs is a substring of a permutation of v_{1}, then w is said to have a regular Abelian period (p, t) where p = |v1| and t = |v_{s}|. If a substring w1[i.i+l-1] of a string w1 and a substring w2[j.j + l - 1] of another string w2 are Abelian equivalent, then the substrings are said to be a common Abelian factor of w1 and w2 and if the length l is the maximum of such then the substrings are said to be a longest common Abelian factor of w1 and w2. We propose efficient algorithms which compute these Abelian regularities using the run length encoding (RLE) of strings. For a given string w of length n whose RLE is of size m, we propose algorithms which compute all Abelian squares occurring in w in O(mn) time, and all regular Abelian periods of w in O(mn) time. For two given strings w1 and w2 of total length n and of total RLE size m, we propose an algorithm which computes all longest common Abelian factors in O(m^{2}n) time.

Original language | English |
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Title of host publication | Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers |

Editors | William F. Smyth, Ljiljana Brankovic, Joe Ryan |

Publisher | Springer Verlag |

Pages | 420-431 |

Number of pages | 12 |

ISBN (Print) | 9783319788241 |

DOIs | |

Publication status | Published - Jan 1 2018 |

Event | 28th International Workshop on Combinational Algorithms, IWOCA 2017 - Newcastle, NSW, Australia Duration: Jul 17 2017 → Jul 21 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10765 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 28th International Workshop on Combinational Algorithms, IWOCA 2017 |
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Country | Australia |

City | Newcastle, NSW |

Period | 7/17/17 → 7/21/17 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers*(pp. 420-431). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10765 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-78825-8_34

**Computing abelian string regularities based on RLE.** / Sugimoto, Shiho; Noda, Naoki; Inenaga, Shunsuke; Bannai, Hideo; Takeda, Masayuki.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10765 LNCS, Springer Verlag, pp. 420-431, 28th International Workshop on Combinational Algorithms, IWOCA 2017, Newcastle, NSW, Australia, 7/17/17. https://doi.org/10.1007/978-3-319-78825-8_34

}

TY - GEN

T1 - Computing abelian string regularities based on RLE

AU - Sugimoto, Shiho

AU - Noda, Naoki

AU - Inenaga, Shunsuke

AU - Bannai, Hideo

AU - Takeda, Masayuki

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Two strings x and y are said to be Abelian equivalent if x is a permutation of y, or vice versa. If a string z satisfies z = xy with x and y being Abelian equivalent, then z is said to be an Abelian square. If a string w can be factorized into a sequence v1, …, vs of strings such that v1, …, vs-1 are all Abelian equivalent and vs is a substring of a permutation of v1, then w is said to have a regular Abelian period (p, t) where p = |v1| and t = |vs|. If a substring w1[i.i+l-1] of a string w1 and a substring w2[j.j + l - 1] of another string w2 are Abelian equivalent, then the substrings are said to be a common Abelian factor of w1 and w2 and if the length l is the maximum of such then the substrings are said to be a longest common Abelian factor of w1 and w2. We propose efficient algorithms which compute these Abelian regularities using the run length encoding (RLE) of strings. For a given string w of length n whose RLE is of size m, we propose algorithms which compute all Abelian squares occurring in w in O(mn) time, and all regular Abelian periods of w in O(mn) time. For two given strings w1 and w2 of total length n and of total RLE size m, we propose an algorithm which computes all longest common Abelian factors in O(m2n) time.

AB - Two strings x and y are said to be Abelian equivalent if x is a permutation of y, or vice versa. If a string z satisfies z = xy with x and y being Abelian equivalent, then z is said to be an Abelian square. If a string w can be factorized into a sequence v1, …, vs of strings such that v1, …, vs-1 are all Abelian equivalent and vs is a substring of a permutation of v1, then w is said to have a regular Abelian period (p, t) where p = |v1| and t = |vs|. If a substring w1[i.i+l-1] of a string w1 and a substring w2[j.j + l - 1] of another string w2 are Abelian equivalent, then the substrings are said to be a common Abelian factor of w1 and w2 and if the length l is the maximum of such then the substrings are said to be a longest common Abelian factor of w1 and w2. We propose efficient algorithms which compute these Abelian regularities using the run length encoding (RLE) of strings. For a given string w of length n whose RLE is of size m, we propose algorithms which compute all Abelian squares occurring in w in O(mn) time, and all regular Abelian periods of w in O(mn) time. For two given strings w1 and w2 of total length n and of total RLE size m, we propose an algorithm which computes all longest common Abelian factors in O(m2n) time.

UR - http://www.scopus.com/inward/record.url?scp=85045999628&partnerID=8YFLogxK

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U2 - 10.1007/978-3-319-78825-8_34

DO - 10.1007/978-3-319-78825-8_34

M3 - Conference contribution

SN - 9783319788241

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 420

EP - 431

BT - Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers

A2 - Smyth, William F.

A2 - Brankovic, Ljiljana

A2 - Ryan, Joe

PB - Springer Verlag

ER -