Although overcoming some limitations of the generate-and-test approach, unimodular transformations are limited to perfect loop nests only. Extending the unimodular approach, this paper describes a framework that enables the use of unimodular transformations to restructure imperfect loop nests. The concepts used previously for perfect loop nests, such as iteration vector, iteration space and lexicographic order, are generalised and some new concepts like preorder tree are introduced. Multiple unimodular transformations are allowed, one each statement in the loop nest. A code generation algorithm is provided that produces a possibly imperfect loop nest to scan an iteration space that is given as a union of sets of affine constraints.
TRACO: An Automatic Loop Nest Parallelizer for Numerical Applications
