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.

• Type: Conference
• Year: 1998
• Where: IPPS
• Authors: Peter R. Cappello, Ömer Egecioglu
• Keywords: Directed Acyclic Graph, Distributed And Parallel Computing, IPPS 1998, Lower Bounds, Precedenceconstrained Multiprocessor Schedules