Symbolic loop parallelization for balancing I/O and memory accesses on processor arrays

Loop parallelization techniques for massively parallel processor arrays using one-level tiling are often either I/O- or memory-bounded, exceeding the target architecture's capabilities. Furthermore, if the number of available processing elements is only known at runtime - as in adaptive systems - static approaches fail. To solve these problems, we present a hybrid compile/runtime technique to symbolically parallelize loop nests with uniform dependences on multiple levels. At compile time, two novel transformations are performed: (a) symbolic hierarchical tiling followed by (b) symbolic multi-level scheduling. By tuning the size of the tiles on multiple levels, a trade-off between the necessary I/O-bandwidth and memory is possible, which facilitates obeying resource constraints. The resulting schedules are symbolic with respect to the number of tiles; thus, the number of processing elements to map onto does not need to be known at compile time. At runtime, when the number is known, a simple prolog chooses a feasible schedule with respect to I/O and memory constraints that is latency-optimal for the chosen tile size. In this way, our approach dynamically chooses latency-optimal and feasible schedules while avoiding expensive re-compilations.