A Compiler for Parallel Exeuction of Numerical Python Programs on ...

solve <strong>for</strong> unknown parameters <strong>of</strong> E using an integer programming solver. The equations arederived as follows:1. E can be constrained such that the m-th component <strong>of</strong> E must be a superset <strong>of</strong>RCSLMAD L m . The idea is to assume that <strong>for</strong> each V ɛD, there exists a point V ′ ɛD ′such that V ′ = V + {t m1 , t m2 , ..., t mn } and that E(m)(V ′ ) = L m (V ). The values t mkare assumed to be unknown integer constants. The following equations can be stated:d∑d∑b m + p k ∗ i k = b ′ m + p k ∗ (i k + t mk ) (5.51)k=1k=1u ′ 1 ≥ u k + t m1 (5.52)u ′ 2 ≥ u 2 + t m2 (5.53).u ′ d ≥ u d + t md (5.54)t m1 ≥ 0 (5.55)t m2 ≥ 0 (5.56).t mk ≥ 0 (5.57)If suitable integer values are found <strong>for</strong> the unknowns t mk and u ′ kthat satisfy the aboveconstraints, then E is a superset <strong>of</strong> the union <strong>of</strong> the RCSLMADs by construction.2. Each component <strong>of</strong> E must be an RCSLMAD and must there<strong>for</strong>e satisfy constraintsrelating the upper bounds and strides.For each integer k such that 2 ≤ k ≤ dd∑p ′ k ≥ p d + (p j ∗ (u ′ j − 1)) (5.58)j=k+13. From the definition <strong>of</strong> an ERL, the difference between any pair <strong>of</strong> bases (b ′ x, b ′ y shouldbe less than stride p d in the last dimension.For {(x, y)|1 ≤ x ≤ n, 1 ≤ y ≤ n, x ≠ y}b x − b y ≤ p d − 1 (5.59)These inequalities <strong>for</strong>ms a set <strong>of</strong> (n − 1) 2 constraints necessary <strong>for</strong> ensuring that E isan ERL.Thus a total <strong>of</strong> n equality constraints and n ∗ d + (d − 1) + (n − 1) 2 inequality constraintscan be derived <strong>for</strong> a total <strong>of</strong> n + d + n ∗ d unknowns. The unknown variables are u ′ k , b′ m45