Affine-by-Statement Scheduling of Uniform and Affine Loop Nests over Parametric Domains

Abstract

This paper deals with parallel scheduling techniques for uniform and affine loop nests. We deal with affine-by-statement scheduling, a powerful extension of Lamport′s hyperplane method where each statement within the loop nest is scheduled by a different timing function. We present a new, constructive and efficient method to determine the optimal (i.e., with smallest latency) affine-by-statement scheduling. We also consider parametric loop nests, where loop limits (in addition to being affine functions of outer loops) involve program variables whose values may not be known at compile-time (but are runtime constants). We then derive parameter-independent affine-by-statement schedules, and we show that these schedules are asymptotically as efficient as parameter-dependent solutions while much more regular. This theoretical result is of importance in practice, as regularity is a key factor for loop rewriting and code generation.