Single-dimension software pipelining for multi-dimensional loops

Traditionally, software pipelining is applied either to the innermost loop of a given loop nest or from the innermost loop to outer loops. We propose a three-step approach, called single-dimension software pipelining (SSP), to software pipeline a loop nest at an arbitrary loop level. The first step identifies the most profitable loop level for software pipelining in terms of initiation rate or data reuse potential. The second step simplifies the multidimensional data-dependence graph (DDG) into a 1-dimensional DDG and constructs a 1-dimensional schedule for the selected loop level. The third step derives a simple mapping function which specifies the schedule time for the operations of the multidimensional loop, based on the 1-dimensional schedule. We prove that the SSP method is correct and at least as efficient as other modulo scheduling methods. We establish the feasibility and correctness of our approach by implementing it on the IA-64 architecture. Experimental results on a small number of loops show significant performance improvements over existing modulo scheduling methods that software pipeline a loop nest from the innermost loop.