Software Pipelining for Nested Loops

In this paper, we present a novel framework of software pipelining for nested loops. Under this framework, a periodic scheduling function, called r-periodic schedule, is associated with each operation of the loop body in the entire iteration space. We present a simple problem formulation as well as e cient solutions which gives provable asymptotically time-optimal schedule for nested loops under our program model. It also provides some useful insights on the interplay between unimodular loop transformation and ne-grain scheduling under a uni ed framework. The main results are: We show that, for nested loops under our program model, software pipelining problem can be formulated into a very simple linear programming problem based on r-periodic schedules. We prove that the r-periodic scheduling is asymptotically time-optimal, that is, it is asymptotically as good as the best schedule | the free schedule where statements can be freely scheduled as early as possible subject only to data dependence constraints. When the nesting level n is small, we present an e cient polynomial time solution method. This surprisingly e cient solution is based on a vectorized extension of the solution of the minimum cost-to-time ratio cycle (MCTRC) problem [29]. It is often desirable to use a small xed periodicity r in the schedule. We present an e cient solution to handle this situation. A discussion of related and future work is included.