On separable and rectangular clusterings
For a finite set of points S in the Euclidean plane we introduce the so called C-s-clustering problem which can be stated as: Partition S into C subsets Si such that Si is separable from S/Si by a line and |Si|=ki, where ki are given numbers. For a function f which maps C subsets Si of S into ℝ we present an algorithm which finds an, with respect to f, optimal C-s-clustering in O(Cn3/2log2n+PC(Cn3/2Uf(n)+Pf(n))) steps. (where Pf(n) resp. Uf(n) are the time to calculate resp. to update f, if the arguments are slightly changed and pC is the number of, for the algorithm distinct, orderings of the ki. These orderings are also a part of the input of the algorithm.) Then an O(nC−1logn) solution for the C-r-clustering problem of finding all sets of C (C≤3) axis-parallel rectangles Ri such that |Ri∩S|=ki and Ri ∩ Rj ∩ S=ø for i ≠ j is given. If we assume in addition that Ri∩Rj=ø for all i ≠ j we give an O(nC-2) algorithm for C>2 and an O(n) algorithm for C=2.
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- 1.AHO, HOPCROFT, ULLMAN, The design and analysis of computer algorithms, Addison Wesley, 1974Google Scholar
- 2.A. ALON, U. ASCHER, Model and solution strategy for placement of rectangular blocks in the Euclidean plane, IEEE Transactions on Computer Aided Design, Vol. 7, No. 3, 1988, pp. 378–386Google Scholar
- 3.F. AURENHAMMER, Power diagrams: properties, algorithms and applications, SIAM J. Comp., Vol. 16, No. 1, 1987, pp. 78–96Google Scholar
- 4.R. COLE, M.SHARIR, C. YAP, On k-hulls and related problems, ACM SIGACT, Symp. on. Theory of Computing, 1984, pp. 154–166Google Scholar
- 5.F. DEHNE, An O(n4) algorithm to construct all Voronoi diagrams for K nearest neighbor searching in the Euclidean plane, Proceedings of the 10th International Colloquium on Automata, Languages and Programming (ICALP '83), Barcelona, Spain, Lecture Notes in Comp. Sci., No. 154, pp. 160–172Google Scholar
- 6.F. DEHNE, H. NOLTEMEIER, Clustering methods for geometric objects and applications to design problems, The Visual Computer, Vol. 2, Springer 1986, pp. 31–38Google Scholar
- 7.F. DEHNE, H. NOLTEMEIER, A computational geometry approach to clustering problems, Proc. 1st ACM Siggraph Symp. Comput. Geom., Baltimore, MD, USA, 1985Google Scholar
- 8.F. DEHNE, H. NOLTEMEIER, Clustering geometric objects and applications to layout problems, Proc. Comput. Graph., Springer Tokyo, 1985Google Scholar
- 10.H. EDELSBRUNNER, E. WELZL, On the number of line separations of a finite set in the plane, Journal of Combinatorial Theory, Vol. 38, No. 1, 1985, pp. 15–29Google Scholar
- 12.D. T. LEE, On k-nearest neighbor Voronoi diagrams in the plane, IEEE Trans. on Comp., Vol. c-31, No. 6, 1982, pp. 478–487Google Scholar
- 13.F. P. PREPARATA, M. I. SHAMOS, Computational geometry, an introduction, Springer New York, 1985Google Scholar
- 14.E. WELZL, More on k-sets of finite sets in the plane, Discrete Comput. Geom., Vol. 1, 1986, pp. 95–100Google Scholar
- 15.F. F. YAO, A 3-space partition and its applications, Proc. 15th ACM Symp. on Theory of Comp., 1983, pp. 258–263Google Scholar