Geometric pattern matching in d-dimensional space

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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 d>3. 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(n 3 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 L 2 metric in d-space in time O(n ⌊3d/2⌋+1 log3 n).