Multi-dimensional Kernel Generation for Loop Nest Software Pipelining

Single-dimension Software Pipelining (SSP) has been proposed as an effective software pipelining technique for multidimensional loops [18]. However, the scheduling methods that actually produce the kernel code have not been published yet. Because of the multi-dimensional nature of SSP kernels, the scheduling problem is more complex and challenging than with modulo scheduling. The scheduler must handle multiple subkernels and initiation rates, scheduling constraints specific to SSP while producing a solution that minimizes the execution time of the final schedule. In this paper three approaches are proposed: the flat method, which schedules operations from different loop levels with the same priority, the level-by-level method, which schedules innermost operations first and does not let other operations interfere with the already scheduled levels, and the hybrid method, which uses the level-by-level mechanism for the innermost level and the flat solution for the other levels. We also break a scheduling constraint introduced in earlier publications and allow for a more compact kernel. The proposed approaches, combined with different sets of heuristics, have been implemented in the Open64/ORC compiler, compared on loop nests from the Livermore and NAS benchmarks.