A parallel algorithm for finding the constrained Voronoi diagram of line segments in the plane

In this paper, we present an O(1/alpha log n) (for any constant 0 less than or equal to alpha less than or equal to 1) time parallel algorithm for constructing the constrained Voronoi diagram of a set L of n non-crossing line segments in E-2, using O(n(1+alpha)) processors on a CREW PRAM model. This parallel algorithm also constructs the constrained Delaunay triangulation of L in the same time and processor bound by the duality. Our method established the conversions from finding the constrained Voronoi diagram L to finding the Voronoi diagram of S, the endpoint set of L. We further showed that this conversion can be done in O(log n) time using n processors in CREW PRAM model. The complexity of the conversion implies that ally improvement of the complexity for finding the Voronoi diagram of a point set will automatically bring the improvement of the one in question.