## 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(loglogn)-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(logn) time. We complement this by showing that O(polylogn) update time implies Ω(logn/loglogn) 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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Pages (from-to) | 66-73 |

Number of pages | 8 |

Journal | Theoretical Computer Science |

Volume | 710 |

DOIs | |

State | Published - 1 Feb 2018 |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier B.V.

## Keywords

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

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science