### Abstract

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O (log n) query time in the (arithmetic) pointer machine model. This improves the previous O log^{3}/^{2} n bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_{w} n + k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.

Original language | English (US) |
---|---|

Title of host publication | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |

Editors | Christos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella |

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

ISBN (Electronic) | 9783959770767 |

DOIs | |

State | Published - Jul 1 2018 |

Event | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic Duration: Jul 9 2018 → Jul 13 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 107 |

ISSN (Print) | 1868-8969 |

### Other

Other | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |
---|---|

Country | Czech Republic |

City | Prague |

Period | 7/9/18 → 7/13/18 |

### Keywords

- Geometric data structures
- I/O model
- Orthogonal point location
- Pointer machines
- Rectangle stabbing
- Word RAM model

### ASJC Scopus subject areas

- Software

## Fingerprint Dive into the research topics of 'Orthogonal point location and rectangle stabbing queries in 3-d'. Together they form a unique fingerprint.

## Cite this

*45th International Colloquium on Automata, Languages, and Programming, ICALP 2018*[31] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 107). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ICALP.2018.31