Efficient parallel geometric algorithms on a mesh of trees

In this paper, we present some efficient parallel geometric algorithms for computing the All Nearest Neighbors, Delaunay Triangulation, Convex Hull, and Voronoi Diagram of a point set S with N points in the plane. The algorithm of All Nearest Neighbors is to find the nearest-neighbor point for each point in S. It can be applied to cluster analysis, classification theory and computational geometry. A Delaunay Triangulation of S is an triangulation in which the circumcircle of each triangle contains no any other point of S. Delaunay Triangulation has practical applications on finite-element method, computational fluid dynamics, geometric modeling, visualization, numerical analysis, and computational geometry. The Convex Hull of S is the smallest convex polygon that includes all the points of S. Convex hull has many applications in pattern recognition, image processing, stock cutting and allocation, and computational geometry. The straight-line dual of a Voronoi Diagram is a Delaunay Triangulation. Voronoi Diagram is a very useful data structure for robotics, image processing, graph theory, computational fluid dynamics, and computational geometry. We use a mesh of trees with N × N processors as the computation model. All of these parallel algorithms have the same good time complexity O (log N) [1][9].