### 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) |
---|---|

Pages (from-to) | 257-274 |

Number of pages | 18 |

Journal | Discrete and Computational Geometry |

Volume | 21 |

Issue number | 2 |

State | Published - 1999 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

*21*(2), 257-274.

**Geometric pattern matching in d-dimensional space.** / Chew, L. P.; Dor, D.; Efrat, Alon; Kedem, K.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 21, no. 2, pp. 257-274.

}

TY - JOUR

T1 - Geometric pattern matching in d-dimensional space

AU - Chew, L. P.

AU - Dor, D.

AU - Efrat, Alon

AU - Kedem, K.

PY - 1999

Y1 - 1999

N2 - 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 log2 n) for 3 < d ≤ 8, and in time O(n5d/4 log2 n) for any d > 8. Thus we improve the previous time bound of O(n2d-2 log2 n) due to Chew and Kedem. For d = 3 we obtain a better result of O(n3 log2 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 L2 metric in d-space in time O(n[3d/2]+1+δ), for any δ > 0.

AB - 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 log2 n) for 3 < d ≤ 8, and in time O(n5d/4 log2 n) for any d > 8. Thus we improve the previous time bound of O(n2d-2 log2 n) due to Chew and Kedem. For d = 3 we obtain a better result of O(n3 log2 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 L2 metric in d-space in time O(n[3d/2]+1+δ), for any δ > 0.

UR - http://www.scopus.com/inward/record.url?scp=0033478042&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033478042&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033478042

VL - 21

SP - 257

EP - 274

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -