Hamiltonian Recurrence for ILP scheduling

Software pipelining has become a very extended loop execution model since microprocessors include more functional units. Scheduling techniques developed for software pipeline a loop have to look for heuristics, thus they achieve the optimal scheduling only in some cases. In this paper we present another software pipelining approach based on a methodology that divides the scheduling problem in two subproblems: one, search for a hamiltonian recurrence and two, generation of parallel code. The first phase is the subject of this paper and holds the NPcompleteness of the problem. The second phase is a solved problem for multiprocessors [ALTLV91] and for microprocessors with ILP [BLAV95]. New edges must be included when no hamiltonian recurrence exists. We present a greedy algorithm for generating such edges and compare it with other software pipelining techniques. Results are given for random graphs and for real graphs. They show that more than 98% of the times an optimal solution is found.