Stabbing Circles for Sets of Segments in the Plane - Claverol, Mercè; Khramtcova, Elena; Papadopoulou, Evanthia; Saumell, Maria; Seara, Carlos (Algorithmica, 2018)
Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to two cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute a representation of all the combinatorially different stabbing circles for S , and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a O ( n log 2 n ) time and O ( n ) space algorithm. We also observe that the stabbing circle problem for S can be solved in worst-case optimal O ( n 2 ) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. Finally we show that the problem of computing the stabbing circle of minimum radius for a set of n parallel segments of equal length has an Ω ( n log n ) lower bound.