Geometric Pattern Matching in D-dimensional Space ? Alon Efrat

We show that, using the L1 metric, the minimum Hausdorr 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 d > 3. 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 diierent translations that achieve the minimum Hausdorr distance in d-space is (n b3d=2c). Furthermore, we present an algorithm which computes the minimum Hausdorr distance under the L2 metric in d-space in time O(n d3d=2e+1 log 3 n).