Processor lower bound formulas for array computations and parametric Diophantine systems

Using a directed acyclic graph (dag) model of algorithms, we solve a problem related to precedence-constrained multiprocessor schedules for array computations. Given a sequence of dags and linear schedules parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n. In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions d_n to a set of parametric linear Diophantine equations. We illustrate an algorithm based on generating functions for constructing a formula for these numbers d_n. The algorithm has been implemented as a Mathematica program. An example run and the symbolic formula for processor lower bounds automatically produced by the algorithm for Gaussian elimination is presented