### Abstract

In this paper we investigate data-structures obtained by a recursive partitioning of the input domain into regions of equal size. One of the most well known examples of such a structure is the quadtree, used here as a basis for more complex data structures; we also provide multidimensional versions of the stratified tree by van Emde Boas. We show that under the assumption that the input points have limited precision (i.e. are drawn from the integer grid of size u) these data structures yield efficient solutions to many important problems. In particular, they allow us to achieve O(log log u) time per operation for dynamic approximate nearest neighbor (under insertions and deletions) and exact on-line closest pair (under insertions only) in any constant dimension. They allow O(log log u) point location in a given planar shape or in its expansion (dilation by a ball of a given radius). Finally, we provide a linear time (optimal) algorithm for computing the expansion of a shape represented by a quadtree. This result shows that the spatial order imposed by this regular data structure is sufficient to optimize the dilation by a ball operation.

Original language | English (US) |
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Pages (from-to) | 160-170 |

Number of pages | 11 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

State | Published - Dec 1 1999 |

Externally published | Yes |

Event | Proceedings of the 1999 IEEE 40th Annual Conference on Foundations of Computer Science - New York, NY, USA Duration: Oct 17 1999 → Oct 19 1999 |

### ASJC Scopus subject areas

- Hardware and Architecture

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## Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*, 160-170.