Efficient nested loop pipelining in high level synthesis using polyhedral bubble insertion

Loop pipelining is a key transformation in high-level synthesis tools as it helps maximizing both computational throughput and hardware utilization. Nevertheless, it somewhat looses its efficiency when dealing with small trip-count inner loops, as the pipeline latency overhead quickly limits its efficiency. Even if it is possible to overcome this limitation by pipelining the execution of a whole loop nest, the applicability of nested loop pipelining has so far been limited to a very narrow subset of loops, namely perfectly nested loops with constant bounds. In this work we propose to extend the applicability of nested-loop pipelining to imperfectly nested loops with affine dependencies by leveraging on the so-called polyhedral model. We show how such loop nest can be analyzed, and under certain conditions, how one can modify the source code in order to allow nested loop pipeline to be applied using a method called polyhedral bubble insertion. We also discuss the implementation of our method in a source-to-source compiler specifically targeted at High-Level Synthesis tools.