Synthesizing systolic arrays with control signals from recurrence equations

Abstract

We present a technique for synthesizing systolic arrays which have non-uniform data flow governed by control signals. The starting point for the synthesis is anAffine Recurrence Equation—a generalization of the simple recurrences encountered in mathematics. A large class of programs, including most (single and multiple) nested-loop programs can be described by such recurrences. In this paper we extend our earlier work (Rajopadhye and Fujimoto 1986) in two principal directions. Firstly, we characterize a class of transformations calleddata pipelining and show that they yield recurrences that havelinear conditional expressions governing the computation. Secondly, we discuss the synthesis of systolic arrays that have non-uniform data flow governed by control signals. We show how to derive the control signals in such arrays by applying similar pipelining transformations to theselinear conditional expressions. The approach is illustrated by deriving the Guibas-Kung-Thompson architecture for computing the cost of optimal string parenthesization.