Parallel-decomposition algorithm for discrete computational problems and its application in developing an efficient discrete convolution algorithm

The authors present a parallel-decomposition algorithm for discrete computational problems. An efficient convolution algorithm is developed under the proposed decomposition algorithm. The decomposition operation is based on integer modular arithmetic. Congruent sets are generated from integer modular operation over the index of the problem and constitute a partition. The partition is used to decompose the problem into several small subproblems. The partition under the integer modular operation is unique. Complete reconstruction of the problem is guaranteed by the uniqueness of the partition. Since the algorithm is established on the foundation of all algebraic systems, i.e. number and set theories, it is highly problem-independent, and provides an uniform approach to parallel decomposition of general problems on the discrete domain. Because the subproblems are highly regular, it is suitable for VLSI hardware implementation. The computational complexity of the developed convolution algorithm is reduced by a factor of p², and (p²)ⁿ for n-dimensional problems (p can be any common factor of the input sequences). Compared to the popular FFT methods, where round-off error is introduced, the convolution result from the algorithm is exact. In addition, it does not have restrictions on the length of the input sequences, as in the well known block convolution algorithm. As a result, the algorithm is highly efficient for convolution of large input sequences or correlation operations