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 present an algorithm based on generating functions for constructing a formula for these numbers d_n. The algorithm has been implemented as a Mathematica program. Example runs and the symbolic formulas for processor lower bounds automatically produced by the algorithm for Matrix-Vector Product, Triangular Matrix Product, and Gaussian Elimination problems are presented. Our approach actually solves the following more general problem: Given an arbitrary r× s integral matrix A and r-dimensional integral vectors b and c, let d_n(n=0,1,…) be the number of solutions in non-negative integers to the system Az=nb+c. Calculate the (rational) generating function ∑_{n≥ 0}d_ntⁿ and construct a formula for d_n.