Complexity of Multi-dimensional Loop Alignment

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 problemcan be stated as a systemof integer linear constraints.

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