Minimal Data Dependence Abstractions for Loop Transformations

Many abstractions of program dependences have already been proposed such as the Dependence Distance the Dependence Di rection Vector the Dependence Level or the Dependence Cone These di erent abstractions have di erent precision The min imal abstraction associated to a transformation is the abstrac tion that contains the minimal amount of information necessary to decide when such a transformation is legal The minimal ab stractions for loop reordering and unimodular transformationsare presented As an example the dependence cone that approxi mates dependences by a convex cone of the dependence distance vectors is the minimal abstraction for unimodular transforma tions It also contains enough information for legally applying all loop reordering transformations and nding the same set of valid mono and multi dimensional linear schedulings than the depen dence distance set