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 3 < d \(\leq\) 8, and in time O(n 5d/4 log 2 n) for any d > 8 . 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 \(\Theta(n^{\floor{3d/2}})\) . Furthermore, we present an algorithm which computes the minimum Hausdorff distance under the L 2 metric in d -space in time \(O(n^{\ceil{3d/2}+1 +\delta})\) , for any δ > 0.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received March 17, 1997, and in revised form January 19, 1998.
Rights and permissions
About this article
Cite this article
Chew, L., Dor, D., Efrat, A. et al. Geometric Pattern Matching in d -Dimensional Space . Discrete Comput Geom 21, 257–274 (1999). https://doi.org/10.1007/PL00009420
Issue Date:
DOI: https://doi.org/10.1007/PL00009420