Time-Optimal and Conflict-Free Mappings of Uniform Dependence Algorithms into Lower Dimensional Processor Arrays

Most existing methods of mapping algorithms into processor arrays are restricted to the case where n-dimensional algorithms or algorithms with n nested loops are mapped into (n—l)-dimensional arrays. However, in practice, it is interesting to map n-dimensional algorithms into (k —l)-dimensional arrays where k\u3c.n. For example, many algorithms at bit-level are at least 4-dimensional (matrix multiplication, convolution, LU decomposition, etc.) and most existing bit level processor arrays are 2-dimensional. A computational conflict occurs if two or more computations of an algorithm are mapped into the same processor and the same execution time. In this paper, necessary and sufficient conditions are derived to identify all mappings without computational conflicts, based on the Hermite normal form of the mapping matrix. These conditions are used to propose methods of mapping any n-dimensional algorithm into (k— l)-dimensional arrays, kn—3, optimality of the mapping is guaranteed