## Abstract

The Lyndon factorization of a string w is a unique factorization ℓ_{1}^{p1},…,ℓ_{m}^{pm} of w such that 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_{1}^{p1},…,ℓ_{m}^{pm} 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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Pages (from-to) | 147-156 |

Number of pages | 10 |

Journal | Theoretical Computer Science |

Volume | 689 |

DOIs | |

Publication status | Published - Aug 15 2017 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)