### Abstract

A word x that is absent from a word y is called minimal if all its proper factors occur in y. Given a collection of k words y-1, y-2,...,y-k over an alphabet Σ, we are asked to compute the set M^ℓ-y-1#...#y-k of minimal absent words of length at most ℓ of word y=y-1#y-2#...#y-k, NotElementΣ. In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. This computation generally requires Ω(n) space for n=|y| using any of the plenty available O(n)-time algorithms. This is because an Ω(n)-sized text index is constructed over y which can be impractical for large n. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when ||M^ℓ-y-1#...#y-N || =o(n), for all N [1, k]. For instance, in the human genome, n ≈ 3 × 10^9 but ||M^12-y-1#...#y-k|| ≈ 10^6. We consider a constant-sized alphabet for stating our results. We show that all M^ℓ-y-1,...,M^ℓ-y-1#...#y-k can be computed in O(kn+k-N=1||M^ℓ-y-1#...#y-N||) total time using O(MaxIn+MaxOut) space, where MaxIn is the length of the longest word in y-1,...,y-k and MaxOut=max{||M^ℓ-y-1#...#y-N||:N [1, k]. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution.

Original language | English |
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Title of host publication | Proceedings - DCC 2019 |

Subtitle of host publication | 2019 Data Compression Conference |

Editors | James A. Storer, Ali Bilgin, Joan Serra-Sagrista, Michael W. Marcellin |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 538-547 |

Number of pages | 10 |

ISBN (Electronic) | 9781728106571 |

DOIs | |

Publication status | Published - 10 May 2019 |

Externally published | Yes |

Event | 2019 Data Compression Conference, DCC 2019 - Snowbird, United States Duration: 26 Mar 2019 → 29 Mar 2019 |

### Publication series

Name | Data Compression Conference Proceedings |
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Volume | 2019-March |

ISSN (Print) | 1068-0314 |

### Conference

Conference | 2019 Data Compression Conference, DCC 2019 |
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Country | United States |

City | Snowbird |

Period | 26/03/19 → 29/03/19 |

### Keywords

- Absent words
- Antidictionaries
- Data compression
- Output sensitive algorithms
- String algorithms

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## Cite this

*Proceedings - DCC 2019: 2019 Data Compression Conference*(pp. 538-547). [8712742] (Data Compression Conference Proceedings; Vol. 2019-March). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/DCC.2019.00062