Automating Non-Unimodular Loop Transformations for Massive Parallelism

Loop transformations have been shown to be very useful in parallelising compilation and regular array design. This paper provides a solution to the open problem of automatic rewriting loop nests for non-unimodular loop transformations. We present an algorithm that rewrites a loop nest under any non-singular (unimodular or non-unimodular) transformation in a mechanical manner. The algorithm works nicely with unimodular transformations being treated as a special case. The extra time complexity incurred due to non-unimodularity is polynomially bounded by the depth of the loop nest.