Novel Data Communication Algorithms on Hypercubes and Related Interconnection Networks and Their Applications in Computational Geometry

We present several novel data communication algorithms for hypercubes. Speci cally, we obtain (1) an algorithm that broadcasts m messages of unit size on a hypercube of size N in optimal time O(m + logN); and (2) algorithms for special cases of computing m pre x sums, also in optimal time O(m + logN). Unlike previous algorithms for performing similar tasks, our schemes require no use of pipelining. They can be implemented using the standard ASCEND/DESCEND strategy commonly used for hypercubes, making their implementations much easier. Moreover, while previous pipelined algorithms require that we know exact embeddings of binary trees into hypercubes, our algorithms use recursive properties of the hypercube. Because of this, our schemes can be easily implemented directly on other similar interconnection networks such as stars and pancakes (both members of the family of recursively decomposable Cayley graphs, to which the hypercube also belongs) without rst having to nd embeddings of tree-like structures of constant degree. To demonstrate the applications of our data communication algorithms, we use them to solve several problems in computational geometry. In particular, we present two parallel algorithms that run in O(m + logN) time, where N is the network size. The rst algorithm locates m planar points in a simple polygon with N vertices. This algorithm works on the hypercube, the star, and the pancake interconnection networks. To our knowledge, this algorithm is the rst for the star and pancake to achieve this performance. For the hypercube, the running time of this algorithm matches that of a previous algorithm (designed for a binary tree) that uses pipelining [3]. The second algorithm locates m planar points in a planar subdivision with O(N) vertices. This algorithm is for the hypercube only. When m = o(logN), it is better than the algorithm by Lee and Preparata [11], designed to locate O(N) points in O(logN) time Work supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)