### Abstract

The Lyndon factorization of a string w is a unique factorization ℓ^{p1}_{1},⋯, ℓ^{pm}_{m} of w s.t. ℓ_{1},⋯, ℓ_{m} is a sequence of Lyndon words that is monotonically decreasing in lexicographic order. In this paper, we consider the reverse-engineering problem on Lyndon factorization: Given a sequence S = ((s_{1}, p_{1}),⋯, (s_{m}, p _{m})) of ordered pairs of positive integers, find a string w whose Lyndon factorization corresponds to the input sequence S, i.e., the Lyndon factorization of w is in a form of ℓ^{p1}_{1},⋯, ℓ^{pm}_{m} with |ℓ_{i}| = s_{i} for all 1 ≤ i ≤ m. Firstly, we show that there exists a simple O(n)-time algorithm if the size of the alphabet is unbounded, where n is the length of the output string. Secondly, we present an O(n)-time algorithm to compute a string over an alphabet of the smallest size. Thirdly, we show how to compute only the size of the smallest alphabet in O(m) time. Fourthly, we give an O(m)-time algorithm to compute an O(m)-size representation of a string over an alphabet of the smallest size. Finally, we propose an efficient algorithm to enumerate all strings whose Lyndon factorizations correspond to S.

Original language | English |
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Title of host publication | Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 565-576 |

Number of pages | 12 |

Edition | PART 2 |

ISBN (Print) | 9783662444641 |

DOIs | |

Publication status | Published - Jan 1 2014 |

Event | 39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 - Budapest, Hungary Duration: Aug 25 2014 → Aug 29 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 8635 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 |
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Country | Hungary |

City | Budapest |

Period | 8/25/14 → 8/29/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings*(PART 2 ed., pp. 565-576). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8635 LNCS, No. PART 2). Springer Verlag. https://doi.org/10.1007/978-3-662-44465-8_48

**Inferring strings from Lyndon factorization.** / Nakashima, Yuto; Okabe, Takashi; Tomohiro, I.; Inenaga, Shunsuke; Bannai, Hideo; Takeda, Masayuki.

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

*Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings.*PART 2 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 2, vol. 8635 LNCS, Springer Verlag, pp. 565-576, 39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014, Budapest, Hungary, 8/25/14. https://doi.org/10.1007/978-3-662-44465-8_48

}

TY - GEN

T1 - Inferring strings from Lyndon factorization

AU - Nakashima, Yuto

AU - Okabe, Takashi

AU - Tomohiro, I.

AU - Inenaga, Shunsuke

AU - Bannai, Hideo

AU - Takeda, Masayuki

PY - 2014/1/1

Y1 - 2014/1/1

N2 - The Lyndon factorization of a string w is a unique factorization ℓp11,⋯, ℓpmm of w s.t. ℓ1,⋯, ℓm is a sequence of Lyndon words that is monotonically decreasing in lexicographic order. In this paper, we consider the reverse-engineering problem on Lyndon factorization: Given a sequence S = ((s1, p1),⋯, (sm, p m)) of ordered pairs of positive integers, find a string w whose Lyndon factorization corresponds to the input sequence S, i.e., the Lyndon factorization of w is in a form of ℓp11,⋯, ℓpmm with |ℓi| = si for all 1 ≤ i ≤ m. Firstly, we show that there exists a simple O(n)-time algorithm if the size of the alphabet is unbounded, where n is the length of the output string. Secondly, we present an O(n)-time algorithm to compute a string over an alphabet of the smallest size. Thirdly, we show how to compute only the size of the smallest alphabet in O(m) time. Fourthly, we give an O(m)-time algorithm to compute an O(m)-size representation of a string over an alphabet of the smallest size. Finally, we propose an efficient algorithm to enumerate all strings whose Lyndon factorizations correspond to S.

AB - The Lyndon factorization of a string w is a unique factorization ℓp11,⋯, ℓpmm of w s.t. ℓ1,⋯, ℓm is a sequence of Lyndon words that is monotonically decreasing in lexicographic order. In this paper, we consider the reverse-engineering problem on Lyndon factorization: Given a sequence S = ((s1, p1),⋯, (sm, p m)) of ordered pairs of positive integers, find a string w whose Lyndon factorization corresponds to the input sequence S, i.e., the Lyndon factorization of w is in a form of ℓp11,⋯, ℓpmm with |ℓi| = si for all 1 ≤ i ≤ m. Firstly, we show that there exists a simple O(n)-time algorithm if the size of the alphabet is unbounded, where n is the length of the output string. Secondly, we present an O(n)-time algorithm to compute a string over an alphabet of the smallest size. Thirdly, we show how to compute only the size of the smallest alphabet in O(m) time. Fourthly, we give an O(m)-time algorithm to compute an O(m)-size representation of a string over an alphabet of the smallest size. Finally, we propose an efficient algorithm to enumerate all strings whose Lyndon factorizations correspond to S.

UR - http://www.scopus.com/inward/record.url?scp=84906230308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906230308&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-44465-8_48

DO - 10.1007/978-3-662-44465-8_48

M3 - Conference contribution

AN - SCOPUS:84906230308

SN - 9783662444641

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

SP - 565

EP - 576

BT - Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings

PB - Springer Verlag

ER -