A Mathematical Formulation of the Loop Pipelining Problem

A mathematical model for the loop pipelining problem is presented. The model considers several parameters for optimization and supports any combination of resource and timing constraints. The unrolling degree of the loop is one of the variables explored by the model. By using Farey’s series, an optimal exploration of the unrolling degree is performed and optimal solutions not considered by other methods are obtained. Finding an optimal schedule that minimizes resource requirements (including registers) is solved by an ILP model. A novel paradigm called branch and prune is proposed to efficiently converge towards the optimal schedule and prune the search tree for integer solutions, thus drastically reducing the running time. This is the first formulation that combines the unrolling degree of the loop with timing and resource constraints in a mathematical model that guarantees optimal solutions.