Three-dimensional orthogonal tile sizing problem : mathematical programming approach

We discuss in this paper the problem of finding the optimal tiling transformation of three-dimensional uniform recurrences on a two-dimensional torus/grid of distributed-memory general-purpose machines. We show that even for the simplest case of recurrences which allows for such transformation, the corresponding problem of minimizing the total running time is a non-trivial non-linear integer programming problem. For the later we derive an O(1) algorithm for finding a good approximation solution. The theoretical evaluations and the experimental results show that the obtained solution approximates the original minimum sufficiently well in the context of the considered problem. Such analytical results are of obvious interest and can be successfully used in parallelizing compilers as well as in performance tuning of parallel codes.