## Abstract

The edit distance (a.k.a. the Levenshtein distance) between two words is defined as the minimum number of insertions, deletions or substitutions of letters needed to transform one word into another. The Levenshtein k-neighbourhood of a word w is the set of words that are at edit distance at most k from w. This is perhaps the most important concept underlying BLAST, a widely-used tool for comparing biological sequences. A natural combinatorial question is to ask for upper and lower bounds on the size of this set. The answer to this question has important algorithmic implications as well. Myers notes that "such bounds would give a tighter characterisation of the running time of the algorithm" behind BLAST. We show that the size of the Levenshtein k-neighbourhood of any word of length n over an arbitrary alphabet is not smaller than the size of the Levenshtein k-neighbourhood of a unary word of length n, thus providing a tight lower bound on the size of the Levenshtein k-neighbourhood. We remark that this result was posed as a conjecture by Dufresne at WCTA 2019.

Original language | English |
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Title of host publication | 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020) |

Editors | Inge Li Gortz, Oren Weimann |

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

Pages | 1-12 |

Number of pages | 12 |

ISBN (Electronic) | 9783959771498 |

DOIs | |

Publication status | Published - 2020 |

Event | 31st Annual Symposium on Combinatorial Pattern Matching, CPM 2020 - Copenhagen, Denmark Duration: 17 Jun 2020 → 19 Jun 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 161 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Annual Symposium on Combinatorial Pattern Matching, CPM 2020 |
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Country/Territory | Denmark |

City | Copenhagen |

Period | 17/06/20 → 19/06/20 |

### Funding

Funding This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 872539. Panagiotis Charalampopoulos: Supported by ERC grant TOTAL under the European Union’s Horizon 2020 Research and Innovation Programme (agreement no. 677651). Jakub Radoszewski: Supported by the Polish National Science Center, grant number 2018/31/D/ST6/03991. Tomasz Waleń: Supported by the Polish National Science Center, grant number 2018/31/D/ST6/03991. Wiktor Zuba: Supported by the Polish National Science Center, grant number 2018/31/D/ST6/03991.

Funders | Funder number |
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Polish National Science Center | 2018/31/D/ST6/03991 |

Horizon 2020 Framework Programme | 677651 |

H2020 Marie Skłodowska-Curie Actions | 872539 |

European Research Council | |

Total |

## Keywords

- Combinatorics on words
- Edit distance
- Levenshtein distance