### Abstract

We show that, using the L∞ metric, the minimum Hausdorff distance under translation between two point sets of cardinality n in d-dimensional space can be computed in time O(n^{(4d-2)/3} log^{2} n) for 3 < d ≤ 8, and in time O(n^{5d/4} log^{2} n) for any d > 8. Thus we improve the previous time bound of O(n^{2d-2} log^{2} n) due to Chew and Kedem. For d = 3 we obtain a better result of O(n^{3} log^{2} n) time by exploiting the fact that the union of n axis-parallel unit cubes can be decomposed into O(n) disjoint axis-parallel boxes. We prove that the number of different translations that achieve the minimum Hausdorff distance in d-space is Θ(n^{[3d/2]}). Furthermore, we present an algorithm which computes the minimum Hausdorff distance under the L_{2} metric in d-space in time O(n^{[3d/2]+1+δ}), for any δ > 0.

Original language | English (US) |
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Pages (from-to) | 257-274 |

Number of pages | 18 |

Journal | Discrete and Computational Geometry |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1999 |

Externally published | Yes |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete and Computational Geometry*,

*21*(2), 257-274. https://doi.org/10.1007/PL00009420