On uniformization of affine dependence algorithms

The authors consider the problem of transforming irregular data dependence structures of algorithms with nested loops into more regular ones. Algorithms under consideration are n-dimensional algorithms (algorithms with n nested loops) with affine dependences where dependences are linear functions of index variables of the loop. Methods are proposed to transform these algorithms into uniform dependence algorithms where dependences are independent of the index variables (constant). Some parallelism might be lost due to making them uniform. The parallelism preserved by the uniformity is measured by (1) the total execution time by the optimal linear schedule which assigns each computation in the algorithm an execution time according to a linear function of the index of the computation and (2) the size of the cone spanned by the dependence vectors after achieving uniformity. The objective of making the dependence uniform is to maximize parallelism preserved by the uniformity or to minimize the number of dependences after uniformity