Parallel Scheduling of Recursively Defined Arrays: Revisited

A restructuring transformation is described which can be used to parallelize recurrence relations. The transformation is based on the hyperplane (or wavefront) method, but extends the applicability of the method to irregularly structured recurrences and introduces an algorithm for solving a restricted class of symbolic linear inequalities which arise from such irregularly-structured recurrences. The algorithm for solving this class of linear inequalities is also of general interest to symbolic computing with systems of linear inequalities. Various sub-problems which arise include deciding systems of linear inequalities, finding the set of extreme points and extreme directions of a convex polyhedron, identifying implicit equalities, and eliminating redundant inequalities. Some approaches to these sub-problems are discussed and a sub-algorithm is given for each.