Abstract

Affine loop transformations, where the iteration domain of each statement is transformed by an affine function, have been used by several authors for program optimization. Some have proposed to apply an additional linear transformation on the combined iteration domains of all statements after the affine transformation, called the ordering phase. The affine transformation itself can, in principle, be decomposed into a linear transformation and a translation. In this paper, we show that it is indeed possible to apply linear transformation, translation and ordering successively and that the translation step can even be performed incrementally. However, we also show that the complexity involved in ensuring a valid translation is much higher than in the absence of a subsequent ordering phase. In the latter case, translation is equivalent to loop fusion with loop shifting. We also discuss several cost functions to be used in a translation step in the presence of an ordering phase and some implications of loop fusion on code generation.

Citations