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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics