Integer affine transformations of parametric ℤ-polytopes and applications to loop nest optimization

Abstract

The polyhedral model is a well-known compiler optimization framework for the analysis and transformation of affine loop nests. We present a new method to solve a difficult geometric operation that is raised by this model: the integer affine transformation of parametric ℤ-polytopes. The result of such a transformation is given by a worst-case exponential union of ℤ-polytopes. We also propose a polynomial algorithm (for fixed dimension), to count points in arbitrary unions of a fixed number of parametric ℤ-polytopes. We implemented these algorithms and compared them to other existing algorithms, for a set of applications to loop nest analysis and optimization.