Geometric pattern matching in d-dimensional space

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(n^5d/4 log² n) for any d > 8. Thus we improve the previous time bound of O(n^2d-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.