Affine-by-Statement Transformations of Imperfectly Nested Loops

A majority of loop restructuring techniques developed so far assume that loops are perfectly nested. The unimodular approach unifies three individual transformations – loop interchange, skewing and reversal – but is still limited to perfect loop nests. This paper outlines a framework that enables the use of unimodular transformations to restructure imperfect loop nests. The concepts previously used for perfect loop nests, such as iteration vector, iteration space and lexicographic order, are generalised to characterise imperfect loop nests. Multiple unimodular transformations are allowed, one on each statement in the loop nests. A code generation algorithm is developed that produces the transformed code by scanning a disjunction of conjunctive affine constraints.