Complexity of Multi-dimensional Loop Alignment

Abstract

Loop alignment is a classical program transformation that can enable the fusion of parallel loops, thereby increasing locality and reducing the number of synchronizations. Although the problem is quite old in the one-dimensional case (i.e., no nested loops), it came back recently — with a multi-dimensional form — when trying to refine parallelization algorithms based on multi-dimensional schedules. The main result of this paper is that, unlike the problem in 1D, finding a multi-dimensional shift of statements that makes an innermost loop parallel is strongly NP-complete. Nevertheless, we identify some polynomially-solvable cases that can occur in practice and we show that the general problem can be stated as a system of integer linear constraints.

Shifting is sometimes called alignment. But we prefer to make a distinction between shifting, a technique that consists in shifting some statements with respect to the loop iterations, and alignment, which is one particular goal achieved by this technique.