## Abstract

We consider the problem of preprocessing two strings S and T, of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i, j in S and positions a, b in T, return the optimal alignment score of S[i..j] and T[a..b]. Let N = mn. We present an oracle with preprocessing time N1+o(1) and space N1+o(1) that answers queries in log2+o(1) N time. In other words, we show that we can efficiently query for the alignment score of every pair of substrings after preprocessing the input for almost the same time it takes to compute just the alignment of S and T. Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS). The best previously known oracle with construction time and size O(N) has slow Ω(√ N) query time [Sakai, TCS 2019], and the one with size N1+o(1) and query time log2+o(1) N (using a planar graph distance oracle) has slow Ω(N3/2) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a √ N factor.

Original language | English |
---|---|

Title of host publication | 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 |

Editors | Nikhil Bansal, Emanuela Merelli, James Worrell |

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

ISBN (Electronic) | 9783959771955 |

DOIs | |

State | Published - 1 Jul 2021 |

Event | 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 - Virtual, Glasgow, United Kingdom Duration: 12 Jul 2021 → 16 Jul 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 198 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 |
---|---|

Country/Territory | United Kingdom |

City | Virtual, Glasgow |

Period | 12/07/21 → 16/07/21 |

### Bibliographical note

Funding Information:Funding Panagiotis Charalampopoulos: Supported by Israel Science Foundation grant 592/17. Shay Mozes: Partially supported by Israel Science Foundation grant 592/17. Oren Weimann: Partially supported by Israel Science Foundation grant 592/17.

Publisher Copyright:

© 2021 Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, and Oren Weimann.

## Keywords

- Edit distance
- Longest common subsequence
- Planar graphs
- Voronoi diagrams

## ASJC Scopus subject areas

- Software