Improvements on geometric pattern matching problems
We consider the following geometric pattern matching problem: find the minimum Hausdorff distance between two point sets under translation with L1 or L∞ as the underlying metric. Huttenlocher, Kedem, and Sharir have shown that this minimum distance can be found by constructing the upper envelope of certain Voronoi surfaces. Further, they show that if the two sets are each of cardinality n then the complexity of the upper envelope of such surfaces is Ω(n3). We examine the question of whether one can get around this cubic lower bound, and show that under the L1 and L∞ metrics, the time to compute the minimum Hausdorff distance between two point sets is On2 log2n).
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- 1.H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl, “Congruence, similarity, and symmetries of geometric objects”, Discrete and Computational Geometry, 3(1988), pp. 237–256.Google Scholar
- 2.P.K. Agarwal, M. Sharir, and S. Toledo, “Applications of parametric searching in geometric optimization”, to appear in Proc. Third ACM-SIAM Symposium on Discrete Algorithms, 1992.Google Scholar
- 3.G.N. Frederickson, “Optimal algorithms for tree partitioning”, Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp 168–177.Google Scholar
- 4.G.N. Frederickson and D.B. Johnson, “Finding kth paths and p-centers by generating and searching good data structures”, Journal of Algorithms, 4(1983), pp 61–80.Google Scholar
- 5.D.P. Huttenlocher and K. Kedem, “Efficiently computing the Hausdorff distance for point sets under translation”, Proceedings of the Sixth ACM Symposium on Computational Geometry, 1990, pp. 340–349.Google Scholar
- 6.D.P. Huttenlocher, G.A. Klanderman, and W.J. Rucklidge, “Comparing Images Using the Hausdorff Distance Under Translation”, Cornell Computer Science Department, TR-91-1211, 1991.Google Scholar
- 7.D.P. Huttenlocher, K. Kedem, and M. Sharir, “The upper envelope of Voronoi surfaces and its applications”, Proceedings of the Seventh ACM Symposium on Computational Geometry, 1991, pp 194–203.Google Scholar
- 8.P.J. Heffernan and S. Schirra, “Approximate decision algorithms for point set congruence”, Saarbrucken Computer Science Department, Germany, Technical Report MPI-I-91-110, 1991.Google Scholar
- 9.K. Kedem, R. Livne, J. Pach, and M. Sharir, “On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles”, Discrete and Computational Geometry, 1(1), 1986, pp. 59–72.Google Scholar
- 10.K. Mehlhorn, Data Structures and Algorithms 3: Multi-Dimensional Searching and Computational Geometry, Springer-Verlag, 1984.Google Scholar
- 11.F.P. Preparata and M.I. Shamos, Computational Geometry, Springer-Verlag, New York, 1985Google Scholar