## Abstract

We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v, c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log log n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log n) time. We complement this by showing that O(polylog n) update time implies Ω(log n/log log n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor/descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.

Original language | English |
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Title of host publication | 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016 |

Editors | Roberto Grossi, Moshe Lewenstein |

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

Pages | 25.1-25.12 |

ISBN (Electronic) | 9783959770125 |

DOIs | |

State | Published - 1 Jun 2016 |

Event | 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016 - Tel Aviv, Israel Duration: 27 Jun 2016 → 29 Jun 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 54 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016 |
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Country/Territory | Israel |

City | Tel Aviv |

Period | 27/06/16 → 29/06/16 |

### Bibliographical note

Funding Information:Partially supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 571/14, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG. Partially supported by Israel Science Foundation grant 794/13.

Publisher Copyright:

© Paweł Gawrychowski, Gad M. Landau, Shay Mozes, and Oren Weimann.

## Keywords

- Marked ancestor
- Nearest colored descendant
- Top-trees
- Vertex-label distance oracles

## ASJC Scopus subject areas

- Software