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 log2n) for d>3. Thus we improve the previous time bound of O(n2d−2 log2n) due to Chew and Kedem. For d=3 we obtain a better result of O(n3 log2n) 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 log3n).

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