Geometric pattern matching in d-dimensional space

Abstract

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).

The work of the first and fourth authors was partly supported by AFOSR Grant AFOSR-91-0328. The first author was also supported by ONR Grant N00014-89-J-1946, by ARPA under ONR contract N00014-88-K-0591, and by the Cornell Theory Center which receives funding from its Corporate Research Institute, NSF, New York State, ARPA, NIH, and IBM Corporation.