An Optimal Algorithm For Computing Visible Nearest Foreign Neighbors Among Colored Line Segments

Given a set S of n colored line segments in IR2 that may intersect only in endpoints. Let c(u) denote the color of a line segment u. is an element of S chosen from chi less than or equal to n different colors. Aline segment v is an element of S is a visible nearest foreign neighbor of u E S if v is a nearest foreign neighbor of u. in S, i.e. c(u) not equal c(v) and no segment with a color different from c(u) is closer to u than v, and if there exist points u' is an element of u and v' is an element of v realizing the distance between u, and v that are visible for each other, i.e. the open segment connecting u' and v' is not intersected by an open line segment in S. We present the first optimal -(n log n) algorithm that computes for each line segment u E S all its visible nearest foreign neighbors. The algorithm finds applications in polygon arrangement analysis, VLSI design rule checking and GIS.