Geometric Pattern Matching in d-Dimensional Space

We show that, using the L 1 metric, the minimum Hausdor 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 di erent translations that achieve the minimum Hausdor distance in d-space is (n b3d=2c ). Furthermore, we present an algorithm which computes the minimum Hausdor distance under the L 2 metric in d-space in time O(n d3d=2e+1+ ), for any > 0.