On Time Optimal Supernode Shape

With the objective of minimizing the total execution time of a parallel program on a distributed memory parallel computer, this paper discusses the selection of an optimal supernode shape of a supernode transformation (also known as tiling). We assume that the communication cost is dominated by the startup penalty and therefore, can be approximated by a constant. We identify three parameters of a supernode transformation: supernode size, relative side lengths, and cutting hyperplane directions. For algorithms with perfectly nested loops and uniform dependencies , we give a closed form expression for an optimal linear schedule vector, and a necessary and suucient condition for optimal relative side lengths. We prove that the total running time is minimized by cutting hyperplane direction matrix whose rows are from the surface of the polar cone of the cone spanned by dependence vectors, also known as tiling cone. The results are derived in continuous space and should for that reason be considered approximate.