### Abstract

Quadtrees are a well-known data structure for representing geometric data in the plane, and naturally generalize to higher dimensions. A basic operation is to expand the tree by splitting a given leaf. A quadtree is smooth if adjacent leaf boxes differ by at most one in height. In this paper, we analyze quadtrees that maintain smoothness with each split operation and also maintain neighbor pointers. Our main result shows that the smooth-split operation has an amortized cost of O(1) time for quadtrees of any fixed dimension D. This bound has exponential dependence on D which we show is unavoidable via a lower bound construction. We additionally give a lower bound construction showing an amortized cost of Ω(log n) for splits in a related quadtree model that does not maintain smoothness.

Original language | English (US) |
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Title of host publication | Algorithm Theory, SWAT 2014 - 14th Scandinavian Symposium and Workshops, Proceedings |

Publisher | Springer Verlag |

Pages | 38-49 |

Number of pages | 12 |

ISBN (Print) | 9783319084039 |

DOIs | |

State | Published - 2014 |

Event | 14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014 - Copenhagen, Denmark Duration: Jul 2 2014 → Jul 4 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8503 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014 |
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Country | Denmark |

City | Copenhagen |

Period | 7/2/14 → 7/4/14 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Algorithm Theory, SWAT 2014 - 14th Scandinavian Symposium and Workshops, Proceedings*(pp. 38-49). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8503 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-08404-6_4