Improved Algorithms for Farthest Colored Voronoi Diagram of Segments

Given 77, 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 + 1)alpha(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 log(2) n). Our algorithm has better performance and is simpler than the best previously known O(n log(3) n)-time algorithm.