# Geometric pattern matching in *d*-dimensional space

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DOI: 10.1007/3-540-60313-1_149

- Cite this paper as:
- Paul Chew L., Dor D., Efrat A., Kedem K. (1995) Geometric pattern matching in
*d*-dimensional space. In: Spirakis P. (eds) Algorithms — ESA '95. ESA 1995. Lecture Notes in Computer Science, vol 979. Springer, Berlin, Heidelberg

## 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} 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 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} log^{3}*n*).

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