Tiling multidimensional iteration spaces for nonshared memory machines

This paper addresses the problem of compiling multiply nested loops for nonshared memory machines. The relatively high communication startup costs in these machines renders frequent communication very expensive. Motivated by this, we present a method of aggregating a number of loop iterations into tiles where the tiles execute atomically a processor executing the iterations belonging to a tile receives all the data it needs before executing any one of the iterations in the tile, executes all the iterations in the tile and then sends the data needed by other processors. Since synchronization is not allowed during the execution of a tile, partitioning the iteration space into tiles must not result in deadlock. We first show the equivalence between the problem of finding partitions and the problem of determining the cone for a given set of dependence vectors. We then present an approach to partitioning the iteration space into deadlock-free tiles so that communication volume is minimized. In addition, we discuss a method to optimize the size of tiles for nested loops for nonshared memory machines. This work differs from other approaches to tiling in that we present a method of optimizing grain size of tiles for nonshared memory machines.