### Abstract

String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set γ ⊆ [1.n] is a k-attractor for a string S ∈ Σ^{n} if and only if every distinct substring of S of length at most k has an occurrence crossing at least one of the positions in γ. Finding the smallest k-attractor is NP-hard for k ≥ 3, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the k-attractor problem to a set-cover instance where the string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a k-attractor in near-optimal time and how to quickly compute exact solutions. For example, we prove that a minimum 3-attractor can be found in O(n) time when |Σ| ∈ O(^{3+ϵ}√log n) for some constant ϵ > 0, despite the problem being NP-hard for large Σ.

Original language | English |
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Title of host publication | Proceedings of 26th European Symposium on Algorithms |

Number of pages | 13 |

Volume | 112 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Publication date | 1 Aug 2018 |

ISBN (Print) | 9783959770811 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

Event | 26th Annual European Symposium on Algorithms - Helsinki, Finland Duration: 20 Aug 2018 → 22 Aug 2018 |

### Conference

Conference | 26th Annual European Symposium on Algorithms |
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Country | Finland |

City | Helsinki |

Period | 20/08/2018 → 22/08/2018 |

### Keywords

- Dictionary compression
- Set cover
- String attractors

### Cite this

*Proceedings of 26th European Symposium on Algorithms*(Vol. 112). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2018.52

}

*Proceedings of 26th European Symposium on Algorithms.*vol. 112, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 26th Annual European Symposium on Algorithms, Helsinki, Finland, 20/08/2018. https://doi.org/10.4230/LIPIcs.ESA.2018.52

**String attractors : Verification and optimization.** / Kempa, Dominik; Policriti, Alberto; Prezza, Nicola; Rotenberg, Eva.

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

TY - GEN

T1 - String attractors

T2 - Verification and optimization

AU - Kempa, Dominik

AU - Policriti, Alberto

AU - Prezza, Nicola

AU - Rotenberg, Eva

PY - 2018/8/1

Y1 - 2018/8/1

N2 - String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set γ ⊆ [1.n] is a k-attractor for a string S ∈ Σn if and only if every distinct substring of S of length at most k has an occurrence crossing at least one of the positions in γ. Finding the smallest k-attractor is NP-hard for k ≥ 3, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the k-attractor problem to a set-cover instance where the string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a k-attractor in near-optimal time and how to quickly compute exact solutions. For example, we prove that a minimum 3-attractor can be found in O(n) time when |Σ| ∈ O(3+ϵ√log n) for some constant ϵ > 0, despite the problem being NP-hard for large Σ.

AB - String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set γ ⊆ [1.n] is a k-attractor for a string S ∈ Σn if and only if every distinct substring of S of length at most k has an occurrence crossing at least one of the positions in γ. Finding the smallest k-attractor is NP-hard for k ≥ 3, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the k-attractor problem to a set-cover instance where the string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a k-attractor in near-optimal time and how to quickly compute exact solutions. For example, we prove that a minimum 3-attractor can be found in O(n) time when |Σ| ∈ O(3+ϵ√log n) for some constant ϵ > 0, despite the problem being NP-hard for large Σ.

KW - Dictionary compression

KW - Set cover

KW - String attractors

U2 - 10.4230/LIPIcs.ESA.2018.52

DO - 10.4230/LIPIcs.ESA.2018.52

M3 - Article in proceedings

SN - 9783959770811

VL - 112

BT - Proceedings of 26th European Symposium on Algorithms

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -