Automatic Processor Lower Bound Formulas for Array Computations

In the directed acyclic graph (dag) model of algorithms, consider the following problem for precedence-constrained multiprocessor schedules for array computations: Given a sequence of dags and linear schedules parameterized by Ò, compute a lower bound on the number of processors required by the schedule as a function of Ò. This problem is formulated so that the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions Ò to a set of parametric linear Diophantine equations. Generating function methods are then used for constructing a formula for the numbers Ò. We implemented this algorithm as a Mathematica program. This paper is an overview of the techniques involved and their applications to well-known schedules for MatrixVector Product, Triangular Matrix Product, and Gaussian Elimination dags. Some example runs and automatically produced symbolic formulas for processor lower bounds by the algorithm are given.

• Type: Conference
• Year: 2002
• Where: ISPAN
• Authors: Peter R. Cappello, Ömer Egecioglu
• Keywords: Directed Acyclic Graph, Distributed And Parallel Computing, ISPAN 2002, Lower Bounds, Precedence-constrained Multiprocessor Schedules