## Abstract

A Dyck sequence is a sequence of opening and closing parentheses (of various types) that is balanced. The Dyck edit distance of a given sequence of parentheses S is the smallest number of edit operations (insertions, deletions, and substitutions) needed to transform S into a Dyck sequence. We consider the threshold Dyck edit distance problem, where the input is a sequence of parentheses S and a positive integer k, and the goal is to compute the Dyck edit distance of S only if the distance is at most k, and otherwise report that the distance is larger than k. Backurs and Onak [PODS'16] showed that the threshold Dyck edit distance problem can be solved in O(n + k16) time. In this work, we design new algorithms for the threshold Dyck edit distance problem which costs O(n + k4.782036) time with high probability or O(n + k4.853059) ) deterministically. Our algorithms combine several new structural properties of the Dyck edit distance problem, a refined algorithm for fast (min, +) matrix product, and a careful modification of ideas used in Valiant's parsing algorithm.

Original language | English |
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Title of host publication | ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 |

Publisher | Association for Computing Machinery |

Pages | 3650-3669 |

Number of pages | 20 |

ISBN (Electronic) | 9781611977073 |

State | Published - 2022 |

Event | 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States Duration: 9 Jan 2022 → 12 Jan 2022 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2022-January |

### Conference

Conference | 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 |
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Country/Territory | United States |

City | Alexander |

Period | 9/01/22 → 12/01/22 |

### Bibliographical note

Publisher Copyright:Copyright © 2022 by SIAM.

### Funding

∗Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel. Supported in part by ISF grants no. 1278/16 and 1926/19, by a BSF grant no. 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064). †University of California, Berkeley, USA. Partly supported by Fulbright Postdoctoral Fellowship. ‡University of California, Berkeley, USA. Partly supported by NSF 1652303, 1909046, and HDR TRIPODS 1934846 grants, and an Alfred P. Sloan Fellowship. §DI/ENS, PSL Research University, France. Partially supported by the grant ANR-20-CE48-0001 from the French National Research Agency (ANR).

Funders | Funder number |
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Alfred P. Sloan Fellowship | ANR-20-CE48-0001 |

National Science Foundation | 1909046, 1652303, 1934846 |

European Commission | |

Agence Nationale de la Recherche | |

United States-Israel Binational Science Foundation | 2018364 |

Israel Science Foundation | 1926/19, 1278/16 |

Horizon 2020 | 683064 |