Parallel computing 2D Voronoi diagrams using untransformed sweepcircles

Abstract

Voronoi diagrams are among the most important data structures in geometric modeling. Among many efficient algorithms for computing 2D Voronoi diagrams, Fortune’s sweepline algorithm (Fortune, 1986 [5]) is popular due to its elegance and simplicity. Dehne and Klein (1987) [8] extended sweepline to sweepcircle and suggested computing a type of transformed Voronoi diagram, which is parallel in nature. However, there is no practical implementation of the sweepcircle algorithm due to the difficulty in representing the transformed edges. This paper presents a new algorithm, called untransformed sweepcircle, for constructing Voronoi diagram in R2. Starting with a degenerate circle (of zero radius) centered at an arbitrary location, as the name suggests, our algorithm sweeps the circle by increasing its radius across the plane. At any time during the sweeping process, each site inside the sweep circle defines an ellipse composing of points equidistant from that point and from the sweep circle. The union of all ellipses forms the beach curve-a star shape inside the sweep circle which divides the portion of the plane within which the Voronoi diagram can be completely determined, regardless of what other points might be outside of the sweep circle. As the sweep circle progresses, the intersection of expanding ellipses defines the Voronoi edges. We show that the sweep line algorithm is the degenerate form of the proposed sweep circle algorithm when the circle center is at infinity, and our algorithm has the same time and space complexity as the sweep line algorithm. Our untransformed sweepcircle algorithm is flexible in allowing multiple circles at arbitrary locations to sweep the domain simultaneously. The parallelized implementation is pretty easy without complicated numerical computation; the most complicated case is nothing but an arc-cosine operation. Furthermore, our algorithm supports the additively weighted Voronoi diagrams of which the Voronoi edges are hyperbolic and straight line segments. We demonstrate the efficacy of our parallel sweep circle algorithm using a GPU.