## Abstract

The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices. The LCE problem asks to preprocess a string into a compact data structure that supports fast LCE queries. In this paper we generalize the LCE problem to trees and suggest a few applications of LCE in trees to tries and XML databases. Given a labeled and rooted tree T of size n, the goal is to preprocess T into a compact data structure that support the following LCE queries between subpaths and subtrees in T. Let v_{1}, v_{2}, w_{1}, and w_{2} be nodes of T such that w_{1} and w_{2} are descendants of v_{1} and v_{2} respectively. - LCEPP(v_{1}, w_{1}, v_{2}, w_{2}): (path-path LCE) return the longest common prefix of the paths v_{1} ⇝ w_{1} and v_{2} ⇝ w_{2}. - LCEPT (v_{1}, w_{1}, v_{2}): (path-tree LCE) return maximal path-path LCE of the path v_{1} ⇝ w_{1} and any path from v_{2} to a descendant leaf. - LCETT (v_{1}, v_{2}): (tree-tree LCE) return a maximal path-path LCE of any pair of paths from v_{1} and v_{2} to descendant leaves. We present the first non-trivial bounds for supporting these queries. For LCEPP queries, we present a linear-space solution with O(log^{∗} n) query time. For LCE_{PT} queries, we present a linear-space solution with O((log log n)^{2}) query time, and complement this with a lower bound showing that any path-tree LCE structure of size O(n polylog(n)) must necessarily use Ω(log log n) time to answer queries. For LCETT queries, we present a time-space trade-off, that given any parameter τ, 1 ≤ τ ≤ n, leads to an O(nτ) space and O(n/τ) query-time solution. This is complemented with a reduction to the set intersection problem implying that a fast linear space solution is not likely to exist.

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

Title of host publication | Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings |

Editors | Ugo Vaccaro, Ely Porat, Ferdinando Cicalese |

Publisher | Springer Verlag |

Pages | 52-64 |

Number of pages | 13 |

ISBN (Print) | 9783319199283 |

DOIs | |

State | Published - 2015 |

Event | 26th Annual Symposium on Combinatorial Pattern Matching, CPM 2015 - Ischia Island, Italy Duration: 29 Jun 2015 → 1 Jul 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 9133 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 26th Annual Symposium on Combinatorial Pattern Matching, CPM 2015 |
---|---|

Country/Territory | Italy |

City | Ischia Island |

Period | 29/06/15 → 1/07/15 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing Switzerland 2015.

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science