## Abstract

A string T of length m is periodic in P of length p if P is a substring of T and T[i]=T[i+p] for all 0≤i≤m−p−1 and m≥2p. The shortest such prefix, P, is called the period of T (i.e., P=T[0..p−1]). In this paper we investigate the period recovery problem. Given a string S of length n, find the primitive period(s) P such that the distance between S and a string T that is periodic in P is below a threshold τ. We consider the period recovery problem over both the Hamming distance and the edit distance. For the Hamming distance case, we present an O(nlogn)-time algorithm, where τ is given as ⌊[Formula presented]⌋ for ϵ>0. For the edit distance case, τ=⌊[Formula presented]⌋ and ϵ>0, we provide an O(n^{4/3})-time algorithm.

Original language | English |
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Pages (from-to) | 2-18 |

Number of pages | 17 |

Journal | Theoretical Computer Science |

Volume | 710 |

DOIs | |

State | Published - 1 Feb 2018 |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier B.V.

### Funding

We thank the anonymous referees for their comments and suggestions. Amihood Amir was partially supported by the Israel Science Foundation grant 571/14 , and grant No. 2014028 from the United States-Israel Binational Science Foundation (BSF). Mika Amit and Gad M. Landau were partially supported by the Israel Science Foundation grant 571/14 , grant No. 2014028 from the United States-Israel Binational Science Foundation (BSF) and DFG . Dina Sokol was partially supported by the United States-Israel Binational Science Foundation (BSF) grant No. 2014028 .

Funders | Funder number |
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Deutsche Forschungsgemeinschaft | |

United States-Israel Binational Science Foundation | |

Israel Science Foundation | 571/14, 2014028 |

## Keywords

- Approximate periodicity
- Edit distance
- Hamming distance
- Period recovery