A novel approach for partitioning iteration spaces with variable densities

Efficient partitioning of parallel loops plays a critical role in high performance and efficient use of multiprocessor systems. Although a significant amount of work has been done in partitioning and scheduling of loops with rectangular iteration spaces, the problem of partitioning non-rectangular iteration spaces --- e.g., triangular, trapezoidal iteration spaces --- with variable densities has not been addressed so far to the best of our knowledge. In this paper, we present a mathematical model for partitioning N-dimensional non-rectangular iteration spaces with variable densities. We present a unimodular loop transformation and a geometric approach for partitioning an iteration space along an axis corresponding to the outermost loop across a given number of processors to achieve near-optimal performance, i.e., to achieve near-optimal load balance across different processors. We present a case study to illustrate the effectiveness of our approach.