Merging free trees in parallel for efficient voronoi diagram construction

This paper describes a new approach for constructing the Voronoi diagram of n points in the plane in parallel. Our approach is based on a divide-and-conquer procedure where we implement the “marry” step by merging forests of free trees (to build the “contour” between the subproblem solutions) in O(log log n) time. This merging procedure is based an a √n -divide-and-merge technique reminiscent of the list-merging approach of Valiant. Our method also involves an optimal parallel method for computing the proximity envelope of a point set with respect to a given line. This structure facilitates the use of our fast mering procedure, for it allows the divide-and-conquer procedure to continue without needing to explicitly remove edges of recursively constructed diagrams that are not part of the final diagram. We use this approach to derive two results regarding the deterministic parallel construction of a Voronoi diagram. Specifically, we show that one can solve the Voronoi diagram problem in O(log n log log n) time and O(n log²n) work (which improves the previous time bound while maintaining the same work bound) or, alternatively, in O(log²n) time and O(n log n) work (which improves the previous work bound while maintaining the same time bound). Our model of computation is the CREW PRAM.