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 byn, 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 solutionsdnto a set of parametric linear Diophantine equations. We illustrate an algorithm based on generating functions for constructing a formula for these numbersdn. 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

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 precedenceconstrained 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 dn to a set of parametric linear Diophantine equations. We illustrate an algorithm based on generating functions for constructing a formula for these numbers dn . 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. 1. Introduction We consider array computations, often referred to as systems of uniform recurrence equations. Parallel execution of uniform recurrence equations ..