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 poinrs and a finite scl of data dcpcndcncc vectors. Usually, the convex hull R of thc index set is a nondcgcnerated convex polytope in 72". In this paper, we show that finding an optimal linear schedule for a uniform dependencc algorithm with an arbilrary bounded convex index set is equivalent lo linding a vector of smallcst norm, where the vcctor norm is defined on a symmelric 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 lime complexity 0(36n3 + 12an2 + 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 lime complexity is bettcr than those of the existing methods.