## Abstract

The edit distance between two rooted ordered trees with n nodes labeled from an alphabet Σ is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. Tree edit distance is a well known generalization of string edit distance. The fastest known algorithm for tree edit distance runs in cubic O(n^{3}) time and is based on a similar dynamic programming solution as string edit distance. In this paper we show that a truly subcubic O(n^{3}-ϵ) time algorithm for tree edit distance is unlikely: For Σ = (n), a truly subcubic algorithm for tree edit distance implies a truly subcubic algorithm for the all pairs shortest paths problem. For Σ = O(1), a truly subcubic algorithm for tree edit distance implies an O(n^{k-ϵ}) algorithm for finding a maximum weight k-clique. Thus, while in terms of upper bounds string edit distance and tree edit distance are highly related, in terms of lower bounds string edit distance exhibits the hardness of the strong exponential time hypothesis [Backurs, Indyk STOC'15] whereas tree edit distance exhibits the hardness of all pairs shortest paths. Our result provides a matching conditional lower bound for one of the last remaining classic dynamic programming problems.

Original language | English |
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Title of host publication | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |

Editors | Artur Czumaj |

Publisher | Association for Computing Machinery |

Pages | 1190-1206 |

Number of pages | 17 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

State | Published - 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States Duration: 7 Jan 2018 → 10 Jan 2018 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 7/01/18 → 10/01/18 |

### Bibliographical note

Publisher Copyright:© Copyright 2018 by SIAM.

## ASJC Scopus subject areas

- Software
- Mathematics (all)