Based on the dimensionality reduction technique and the solution for proximate points problem, we achieve the optimality of the three-dimensional Euclidean distance transform (3D_EDT) computation. For an N N N binary image, our algorithms for both 3D_EDT and its applications can be performed in O (log log N) time using CRCW processors or in O (log N) time using EREW processors. To the best of our knowledge, all results described above are the best known. As for the n-dimensional Euclidean distance transform (nD_EDT) and its applications of a binary image of size Nn, all of them can be computed in O (nlog log N) time using CRCW processors or in O (nlog N) time using EREW processors.

The mesh-connected computers with hyperbus broadcasting are an extension of the mesh-connected computers with multiple broadcasting. Instead of using local buses, we use global buses to connect processors. Such a strategy efficiently reduces the time complexity of the semigroup problem from O(N) to O(log N). Also, the matrix multiplication and the transitive closure problems are solved in O(log N) and O(log2 N) time, respectively. Then, based on these operations, several interesting problems such as the connected recognition problem, the articulation problem, the dominator problem, the bridge problem, the sorting problem, the minimum spanning tree problem and the bipartite graph recognition problem can be solved in the order of polylogarithmic time.