An optimal algorithm for constructing the delaunay triangulation of a set of line segments

Abstract

In this paper, we first define a new Voronoi diagram for the endpoints of a set of line segments in the plane which do not intersect (except possibly at their endpoints), which is called a bounded Voronoi diagram . In this Voronoi diagram, the line segments themselves are regarded as obstacles. We present an optimal Θ(n log n) algorithm to construct it, where n is the number of input line segments. We (then show that the straight-line dual of the bounded Voronoi diagram of a set of non-intersecting line segments is the Delaunay triangulation of that set. And the straight-line dual can be obtained in time proportional to the number of input line segments when the corresponding bounded Voronoi diagram is available. Consequently, we obtain an optimal Θ(n log n) algorithm to construct the Delaunay triangulation of a set of n non-intersecting line segments in the plane. Our algorithm improves the time bound O(n2) of the previous best algorithm.