### Abstract

We give a new characterization of maximal repetitions (or runs) in strings, using a tree defined on recursive standard factorizations of Lyndon words, called the Lyndon tree. The characterization leads to a remarkably simple novel proof of the linearity of the maximum number of runs p(n) in a string of length n. Furthermore, we show an upper bound of p(n) < 1.5n, which improves on the best upper bound 1.6n (Crochemore & Hie 2008) that does not rely on computational verification. The proof also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string.

Original language | English |
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Title of host publication | Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |

Publisher | Association for Computing Machinery |

Pages | 562-571 |

Number of pages | 10 |

Edition | January |

ISBN (Electronic) | 9781611973747 |

Publication status | Published - Jan 1 2015 |

Event | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States Duration: Jan 4 2015 → Jan 6 2015 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Number | January |

Volume | 2015-January |

### Other

Other | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |
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Country | United States |

City | San Diego |

Period | 1/4/15 → 1/6/15 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015*(January ed., pp. 562-571). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 2015-January, No. January). Association for Computing Machinery.

**A new characterization of maximal repetitions by Lyndon trees.** / Bannai, Hideo; I, Tomohiro; Inenaga, Shunsuke; Nakashima, Yuto; Takeda, Masayuki; Tsuruta, Kazuya.

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

*Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015.*January edn, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, no. January, vol. 2015-January, Association for Computing Machinery, pp. 562-571, 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, United States, 1/4/15.

}

TY - GEN

T1 - A new characterization of maximal repetitions by Lyndon trees

AU - Bannai, Hideo

AU - I, Tomohiro

AU - Inenaga, Shunsuke

AU - Nakashima, Yuto

AU - Takeda, Masayuki

AU - Tsuruta, Kazuya

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We give a new characterization of maximal repetitions (or runs) in strings, using a tree defined on recursive standard factorizations of Lyndon words, called the Lyndon tree. The characterization leads to a remarkably simple novel proof of the linearity of the maximum number of runs p(n) in a string of length n. Furthermore, we show an upper bound of p(n) < 1.5n, which improves on the best upper bound 1.6n (Crochemore & Hie 2008) that does not rely on computational verification. The proof also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string.

AB - We give a new characterization of maximal repetitions (or runs) in strings, using a tree defined on recursive standard factorizations of Lyndon words, called the Lyndon tree. The characterization leads to a remarkably simple novel proof of the linearity of the maximum number of runs p(n) in a string of length n. Furthermore, we show an upper bound of p(n) < 1.5n, which improves on the best upper bound 1.6n (Crochemore & Hie 2008) that does not rely on computational verification. The proof also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string.

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UR - http://www.scopus.com/inward/citedby.url?scp=84938221386&partnerID=8YFLogxK

M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 562

EP - 571

BT - Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015

PB - Association for Computing Machinery

ER -