Selecting Affine Mappings Based on Performance Estimation

In previous work, we presented a framework for unifying iteration reordering transformations such as loop interchange, loop distribution, loop skewing and statement reordering. The framework provides a uniform way to represent and reason about transformations. However, it does not provide a way to decide which transformation(s) should be applied to a given program. This paper describes a way to make such decisions within the context of the framework. The framework is based on the idea that a transformation can be represented as an affine mapping from the original iteration space to a new iteration space. We show how we can estimate the performance of a program by considering only the mapping from which it was produced. We also show how to produce a lower bound on performance given only a partially specified mapping. Our ability to estimate performance directly from mappings and to do so even for partially specified mappings allows us to efficiently find mappings which will produce good code.