Optimal point location in a monotone subdivision
SIAM Journal on Computing
The upper envelope of Voronoi surfaces and its applications
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Farthest line segment Voronoi diagrams
Information Processing Letters
Farthest-polygon Voronoi diagrams
ESA'07 Proceedings of the 15th annual European conference on Algorithms
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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) log n) 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(n log2 n). Our algorithm has better performance and is simpler than the best previously known O(n log2 n)-time algorithm.