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

The algorithm is very conservative and transfers all memory locations in the memory intervalspanned by the group. The algorithm is correct but transfers too much data and uses toomuch GPU on-board memory. Better solutions are needed in most cases.Algorithm 3 computes a superset <strong>of</strong> union <strong>of</strong> arbitrary RCSLMADsInputs: A group {L 1 , L 2 , .., L n } <strong>of</strong> RCSLMADs defined over domain D.Outputs: Set M, size S <strong>of</strong> set M and function F <strong>for</strong> mapping CPU memory locations toGPU memory locations.1: Compute m 1 = min(min(L 1 (V ), V ɛD), min(L 2 (V ), V ɛD), .., min(L n (V ), V ɛD)).2: Compute m 2 = max(max(L 1 (V ), V ɛD), max(L 2 (V ), V ɛD), .., max(L n (V ), V ɛD)).3: Compute M = {m|m 1 ≤ m ≤ m 2 , mɛZ}.4: Compute S = m 2 − m 1 .5: Construct F (m) = m − m 1 and R = {m|0 ≤ m ≤ m 2 − m 1 , mɛZ}.6: return M, S, and F .While the algorithms 1, 2 and 3 have only been described <strong>for</strong> RCSLMADs, they areactually applicable on a slightly wider class <strong>of</strong> LMADs. For convenience, assume thatthe ordering function is the identity function and consider the condition p k > p k+1 − 1 +d∑p j ∗ (u j − 1) imposed on RCSLMADs. If the condition is instead loosened to p k >j=k+1d∑j=k+1p j ∗ (u j − 1), then the algorithms 1, 2 and 3 are still valid (and can be verified bysubstituting the loosened condition in the corresponding pro<strong>of</strong>s).5.3 More efficient solutions in specific casesIn some cases, it is possible to derive more efficient solutions that transfer less data thanthe general algorithm presented earlier. Computing and reasoning with unions <strong>of</strong> arbitraryRCSLMADs is non-trivial. Consider a group <strong>of</strong> n RCSLMADs. Let p jk be the stride inthe j-th RCSLMAD in k-th dimension. Instead <strong>of</strong> attempting to compute the union <strong>of</strong>RCSLMADs in arbitrary cases, this section is limited to the case p t1k = p t2k, 1 ≤ t 1 , t 2 ≤ n,i.e. all RCSLMADs have the same stride in any given dimension k. Such cases occur whena programmer is accessing multiple array locations in the loop body with fixed distancebetween the array accesses. All the RCSLMADs must have the same ordering functionbecause all the RCSLMADs share strides and are defined over the same domain. Withoutloss <strong>of</strong> generality, assume that the ordering function is identity throughout this section.One example <strong>of</strong> the types <strong>of</strong> problems being studied in this section is as follows:L 1 = 0 + 20 ∗ i + 3 ∗ j (5.23)L 2 = 21 + 20 ∗ i + 3 ∗ j (5.24)0 ≤ i < 5 (5.25)40