An optimal algorithm for computing visible nearest foreign neighbors among colored line segments

Abstract

Given a set S of n colored line segments in ℝ2 that may intersect only in endpoints. Let c(u) denote the color of a line segment u ∃ S chosen from χ ≤ n different colors. A line segment v ∃ S is a visible nearest foreign neighbor of u ∃ S if v is a nearest foreign neighbor of u in S, i.e. c(u) ≠ c(v) and no segment with a color different from c(u) is closer to u than v, and if there exist points u' ∃ u and v' ∃ 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 θ(n log n) algorithm that computes for each line segment u ∃ S all its visible nearest foreign neighbors. The algorithm finds applications in polygon arrangement analysis, VLSI design rule checking and GIS.