Generalising the unimodular approach [program code transformation]

Most of the available parallelism in source code is contained in loops and is exploited by applying a sequence of loop transformations. Different methods of representing and ordering sequences of transformations have been developed, including the use of unimodular transformations, which unify loop permutation, loop reversal, and loop skewing of perfectly nested loops. This paper presents three extensions to the unimodular approach that make it applicable to a wider range of source code structures. First, the unimodular transformations are extended to represent additional loop transformation techniques, namely loop fission, loop fusion, loop blocking (tiling), strip mining, cycle shrinking, loop coalescing, and loop collapsing. Second, the application of unimodular transformations is generalized to handle both perfectly and imperfectly nested loops. Third, attractive properties of the original unimodular transformations are preserved by the generalized model.