Full parallelism in uniform nested loops using multi-dimensional retiming

Most scientific and DSP applications are recursive or iterative. Uniform nested loops can be modeled as multi-dimensional data flow graphs (DFGs). To achieve full parallelism of the loop body, i.e., all the computational nodes executed in parallel, substantially decreases the overall computation time. It is well known that for one-dimensional DFGs retiming can not always achieve full parallelism. This paper shows an important and counter-intuitive result, which proves that we can always obtain full-parallelism for DFGs with more than one dimension. It also presents two novel multi-dimensional retiming techniques to obtain full parallelism.