A NEW APPROACH TO FINDING OPTIMAL LINEAR SCHEDULES FOR UNIFORM DEPENDENCE ALGORITHMS†

Abstract

A uniform dependence algorithm can be represented by an index set of index points and a finite set of data dependence vectors. Usually, the convex hull R of the index set is a nondegenerated convex polytope in R n, In this paper, we show that finding an optimal linear schedule for a uniform dependence algorithm with an arbitrary bounded convex index set is equivalent to finding a vector of smallest norm, where the vector norm is defined on a symmetric convex set R∗ (which is the dual of difference body R − R). A linear programming problem is derived for finding the smallest vector. This problem can be solved in the empirical average time complexity O(36n 3 + 12an 2 + a), where a is the number of the linear inequalities defining the convex hull R and n is the dimension of the index set. This time complexity is better than those of the existing methods.