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http://dx.doi.org/10.5626/JOK.2017.44.3.246

An Algorithm for Computing a Minimum-Width Color-Spanning Rectangular Annulus  

Bae, Sang Won (Kyonggi Univ.)
Publication Information
Journal of KIISE / v.44, no.3, 2017 , pp. 246-252 More about this Journal
Abstract
In this paper, we present an algorithm that computes a minimum-width color spanning axis-parallel rectangular annulus. A rectangular annulus is a closed region between a rectangle and its offset, and it is thus bounded by two rectangles called its outer and inner rectangles. The width of a rectangular annulus is determined by the distance between its outer and inner rectangles. Given n points in the plane each of which has one of the prescribed k colors, we call a rectangular annulus color spanning if it contains at least one point for each of the k colors. Prior to this work, there was no known exact algorithm that computes a minimum-width color-spanning rectangular annulus. Our algorithm is the first to solve this problem and it runs efficiently in $O((n-k)^3nlogn)$ time.
Keywords
color-spanning object; annulus; rectangle; covering problem; computational geometry; algorithm;
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1 F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction, Springer Verlag, 1985.
2 M. de Berg, O. Cheong, M. van Kreveld and M. Overmars, Computationsl Geometry: Alogorithms and Applications, 3rd Ed., Springer-Verlag, 2008.
3 M. Abellanas, F. Hurtado, Icking C., R. Klein, E. Langetepe, L. Ma, P. Palop and V. Sacristan, "Smallest Color-Spanning Objects," Proc. Euro. Sympos. Algo. (ESA 2001), LNCS, Vol. 2161, pp. 278-289, 2001.
4 D. Huttenlocher, K. Kedem and M. Sharir, "The Upper Envelope of Voronoi Surfaces and Its Applications," Discrete Comput. Geom., Vol. 9, pp. 267-291, 1993.   DOI
5 M. Abellanas, F. Hurtado, C. Icking, R. Klein, E. Langetepe, L. Ma, P. Palop and V. Sacristan, "The Farthest Color Voronoi Diagram and Related Problems," Proc. 17th Euro. Comput. Geom (EuroCG 2001), pp. 113-116, 2001.
6 S. Das, P. Goswami and S. Nandy, "Smallest Color-Spanning Object Revisited," Int. J. Comput. Geom. Appl., Vol. 19, pp. 457-478, 2009.   DOI
7 A. D. Wainstein, "A non-monotonous placement problem in the plane," Software Systems for Solving Optimal Planning Problems, Abstract: 9th All-Union Symp. USSR, Symp., 1986.
8 H. Ebara, N. Fukuyama, H. Nakano and Y. Nakanishi, "Roundness algorithms using the Voronoi diagrams," Abstracts 1st Canadian Conf. Comput. Geom (CCCG), 1989.
9 P. K. Agarwal and M. Sharir, "Efficient randomized algorithms for some geometric optimization problems," Discrete Comput. Geom., Vol. 16, pp. 317-337, 1996.   DOI
10 M. Abellanas, F. Hurtado, C. Icking, L. Ma, B. Palop and P. A. Ramos, "Best Fitting Rectangles," Proc. 19th Euro. Workshop Comput. Geom. (EuroCG 2003), 2003.
11 O. Gluckshenko, H. W. Hamacher and A. Tamir, "An optimal O(n log n) algorithm for finding an enclosing planar rectilinear annulus of minimum width," Operations Research Lett., Vol. 37, pp. 168-170, 2009.   DOI
12 S.W. Bae, "Computing a Minimum-Width Square Annulus in Arbitrary Orientation," Proc. 10th Int. Workshop on Algo. Comput. (WALCOM 2016), LNCS, Vol. 9627, pp. 131-142, 2016.
13 J. Mukherjee, P. R. S. Mahapatra, A. Karmakar and S. Das, "Minimum-width rectangular annulus," Theoretical Comput. Sci., Vol. 508, pp. 74-80, 2013.   DOI
14 A. Acharyya, S. Nandy and S. Roy, "Minimum Width Color Spanning Annulus," Proc. 22nd Int. Comput. Combinat. Conf. (COCOON 2016), LNCS, Vol. 9797, pp. 431-442, 2016.
15 J. Hershberger, "Finding the upper envelope of n line segments in O(n log n) time," Inform. Proc. Lett., Vol. 33, pp. 169-174, 1989.   DOI