Abstract
Given n line segments in the plane with each colored by one of k colors, the Farthest Colored Voronoi Diagram (FCVD) is a subdivision of the plane such that the region of a c-colored site (segment or subsegment) s contains all points of the plane for which c is the farthest color and s is the nearest c-colored site. FCVD is a generalization of the Farthest Voronoi Diagram (i.e., k = n) and the regular Voronoi Diagram (i.e., k = 1). In this paper, we first present a simple algorithm to solve the general FCVD problem in an output-sensitive fashion in O((kn + I)α(H)logn) time, where I is the number of intersections of the input and H is the complexity of the FCVD. We then focus on a special case, called Farthest-polygon Voronoi Diagram (FPVD), in which all colored segments form k disjoint polygonal structures (i.e., simple polygonal curves or polygons) with each consisting of segments with the same color. For FPVD, we present an improved algorithm with a running time of O(nlog2 n). Our algorithm has better performance and is simpler than the best previously known O(nlog3 n)-time algorithm.
This research was partially supported by NSF through CAREER Award CCF-0546509 and grant IIS-0713489.
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References
Agarwal, K.P., Sharir, M.: Algorithmic techniques for geometric optimization (1995)
Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment voronoi diagrams. Inf. Process. Lett. 100(6), 220–225 (2006)
Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.-S.: Farthest-polygon voronoi diagrams. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 407–418. Springer, Heidelberg (2007)
de Berg, M., Van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational geometry: algorithms and applications, 2nd edn. (2000)
Edelsbrunner, H., Guibas, L.J., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM J. Comput. 15, 317–340 (1986)
Huttenlocher, D.P., Kedem, K., Sharir, M.: The upper envelope of voronoi surfaces and its applications. In: SCG 1991: Proceedings of the seventh annual symposium on Computational geometry, pp. 194–203. ACM, New York (1991)
Nielsen, F., Yvinec, M.: An output sensitive convex hull algorithm for planar objects. International Journal of Computational Geometry and Applications 8, 39–65 (1995)
Yap, C.K.: An o(n logn) algorithm for the voronoi diagram of a set of simple curve segments. In: Discrete and computational geometry, vol. 2, pp. 365–393 (1987)
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Zhu, Y., Xu, J. (2011). Improved Algorithms for Farthest Colored Voronoi Diagram of Segments. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_29
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DOI: https://doi.org/10.1007/978-3-642-22616-8_29
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