Parallel execution of nested loops in band parallelism

How to execute a nested loop in band parallelism on a multiprocessor system is addressed. The mathematical models of the waveband method, the hyperplane method, the modified hyperplane method and the linear band method are derived and compared. Since the structures of the real multiprocessor systems are at most 3-dimensional, in order to map the loop into these systems, an efficient algorithm for finding the optimal linear band in 2-dimensional index space, instead of a high dimensional index space, is proposed.