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

the smallest set E because any value <strong>of</strong> b 0 smaller than b 1 will result in either the same ora larger domain D ′ .The methodology is generalized by construction in algorithm 4 <strong>for</strong> n RCSLMADs withb 0 = b 1 . The idea is that if the number <strong>of</strong> non-distinct bases in the construct ERL E is q,and if the upper bound <strong>of</strong> the domain <strong>of</strong> E is t n +u, then the total number <strong>of</strong> elements to betransferred to the GPU is (t n +u)∗q since t n +u elements need to be transferred <strong>for</strong> each nondistinctRCSLMAD present in the ERL E. Thus on the GPU, we can allocate space equalto (t n + u) ∗ q elements. For address mapping, consider the interval b 0 to b 0 + m − 1 on thesystem RAM. Out <strong>of</strong> this interval, q elements accessed by the q non-overlapping RCSLMADsare transferred. This pattern <strong>of</strong> q out <strong>of</strong> every m elements is repeated. There<strong>for</strong>e if elementsnot accessed by the ERL were to be discarded, then the pattern is equivalent to q interleavedaccesses with stride q.Algorithm 4 computes the approximate union <strong>of</strong> n one-dimensional RCSLMADs with thesame stride.Inputs: n one-dimensional RCSLMADs {L 1 , L 2 , .., L n } with stride m and bases{b 1 , b 2 , .., b n } such that b 1 ≤ b 2 ... ≤ b n defined over domain D = {i, 0 ≤ i < u}.Outputs: ERL E and a list <strong>of</strong> n expressions representing the trans<strong>for</strong>med address <strong>of</strong> eachRCSLMAD.1: <strong>for</strong> each RCSLMAD L j do2: Compute the term t j = ⌊(b j − b 1 )/m⌋ and r j = (b j − b 1 )%m.3: end <strong>for</strong>4: Construct a list R = {b 1 + r 1 , b 1 + r 2 , .., b n + r n }.5: Remove any duplicates from R. Let the number <strong>of</strong> elements remaining in R be q.6: Sort R in-place.7: Construct a one-dimensional ERL E with stride m, bases R1 and domain D ′ = {j, 0 ≤j ≤ t n + u}.8: Construct an empty list I.9: <strong>for</strong> each RCSLMAD L j do10: Find x such that R(x) = b j + r j .11: Append the expression (represented as an AST or other compiler IR) q ∗ (i + t j ) + xto I where i is the original loop counter <strong>for</strong> which the analysis is being conducted.12: end <strong>for</strong>13: return E and I.5.3.3 Multidimensional RCSLMADs with common stridesConsider n RCSLMADs {L 1 , L 2 , .., L n } with the common strides P = {p 1 , p 2 , .., p d }. Letthe RCSLMADs be defined over a d dimensional domain D = {(i 1 , i 2 , .., i d ) | 0 ≤ i j