An improved parallel algorithm for constructing voronoi diagram on a mesh-connected computer

While constructing a Voronoi diagram V_P for a set P of n points on a mesh-connected computer (MCC), it is necessary to find a set B of edges which are intersected by the dividing chain C during the merge process of two Voronoi diagrams V_L and V_R, where L and R contain the leftmost [n/2] points and the rightmost [n/2] points of P respectively. The computation of B requires two operations: First decide for each edge e in V_L and V_R whether its end vertices are closer to L or R, and then from that information, determine whether e is intersected by C. However, in the previous parallel algorithm each of the former and latter operations requires planar point location which takes O(√n) time on a √n × √n MCC, and in addition the former operation needs to compute convex hulls of L and R. In this paper, we shall show that the latter operation can be done in O(1) time without executing planar point location and the former operation can be executed without the computation of convex hulls. Therefore, the computation of B is reduced to only one planar point location.