An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM Model

In this paper, we develop a parallel algorithm for the 2D Euclidean distance transform (2D_EDT, for short) of a binary image of size N x N in O(1)time using N^{2+(delta)+(epsilon)} CRCW processors and a parallel algorithm for the 3D Euclidean distance transform (3D_EDT, for short) of a binary image of size N x N x N in O(1)time using N^{3+(delta)+(epsilon)} CRCW processors, where (delta)= {1\over k}, (epsilon)= {1\over 2^{c+1}-1}, k, and c are constants and positive integers. Our 2D_EDT (3D_EDT) parallel algorithm can be used to build up Voronoi diagram and Voronoi polygons (polyhedra) in a 2D (3D) binary image also. All of these parallel algorithms can be performed in O(1) time using N^{2+(delta)+(epsilon)} (N^{3+ (delta)+(epsilon)}) CRCW processors. To the best of our knowledge, all results derived above are the best O(1) time algorithms known.