### Abstract

In colored range searching, we are given a set of n colored points in d ≥ 2 dimensions to store, and want to support orthogonal range queries taking colors into account. In the colored range counting problem, a query must report the number of distinct colors found in the query range, while an answer to the colored range reporting problem must report the distinct colors in the query range.
We give the first linear space data structure for both problems in two dimensions (d = 2) with o(n) worst case query time. We also give the first data structure obtaining almost-linear space usage and o(n) worst case query time for points in d > 2 dimensions. Finally, we present the first dynamic solution to both counting and reporting with o(n) query time for d ≥ 2 and d ≥ 3 dimensions, respectively.

Original language | English |
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Title of host publication | Algorithm Theory – SWAT 2014 : Proceedings of the 14th Scandinavian Symposium and Workshops on Algorithm Theory |

Editors | R. Ravi, Inge Li Gørtz |

Publisher | Springer |

Publication date | 2014 |

Pages | 229-240 |

ISBN (Print) | 978-3-319-08403-9 |

ISBN (Electronic) | 978-3-319-08404-6 |

DOIs | |

Publication status | Published - 2014 |

Event | 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2014) - Copenhagen, Denmark Duration: 2 Jul 2014 → 4 Jul 2014 Conference number: 14 http://algolog.compute.dtu.dk/swat2014/ |

### Conference

Conference | 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2014) |
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Number | 14 |

Country | Denmark |

City | Copenhagen |

Period | 02/07/2014 → 04/07/2014 |

Internet address |

Series | Lecture Notes in Computer Science |
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Volume | 8503 |

ISSN | 0302-9743 |

## Cite this

Grossi, R., & Vind, S. J. (2014). Colored Range Searching in Linear Space. In R. Ravi, & I. L. Gørtz (Eds.),

*Algorithm Theory – SWAT 2014: Proceedings of the 14th Scandinavian Symposium and Workshops on Algorithm Theory*(pp. 229-240). Springer. Lecture Notes in Computer Science, Vol.. 8503 https://doi.org/10.1007/978-3-319-08404-6_20