Communication-Minimal Tiling of Uniform Dependence Loops

Tiling is a loop transformation that the compiler uses to create automatically blocked algorithms in order to improve the benefits of the memory hierarchy and reduce the communication overhead between processors. Motivated by existing results, this paper presents a conceptually simple approach to finding tilings with a minimal amount of communication between tiles. The development of all results is based primarily on the inequality of arithmetic and geometric means, except for Lemma 8 whose proof relies on the concept of extremal rays of convex cones. The key insight is mat a tiling that is communication-minimal must induce the same amount of communication through all faces of a tile, which restricts the search space for optimal tilings to those tiling matrices whose rows are all extremal rays in a cone. For nested loops with several special forms of dependences, closed-form optimal tilings are derived. In the general case, a procedure is given that always returns optimal tilings. A detailed comparison of this work with some existing results is provided.