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 an Affine Recurrence Equation&emdash;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 [RFc] in two principal directions. Firstly, we present a class of transformations called data pipelining and show that they yield recurrences that have linear 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 these linear conditional expressions. The approach is illustrated by deriving the Guibas-Kung-Thompson architecture for computing the cost of optimal string parenthesization.