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

1 <strong>for</strong> i 1 in prange(u 1 ) :2 <strong>for</strong> i 2 in prange(u 2 ) :3 . . .4 <strong>for</strong> i m in prange(u m ) :5 <strong>for</strong> i m+1 in range(u m+1 ) :6 .7 .8 <strong>for</strong> i d in range(u d )9 #loop bodyFigure 5.1: Loop nests considered <strong>for</strong> array access analysisloop counters to one-dimensional space <strong>of</strong> memory locations. The space <strong>of</strong> memory locationsrepresents the locations in virtual memory. This space starts at zero and ends at positiveinfinity. The memory space is assumed to consist <strong>of</strong> discrete cells where each cell is composed<strong>of</strong> one discrete unit <strong>of</strong> memory. If all the arrays in the loop have the same element size s,(<strong>for</strong> example if all arrays are arrays <strong>of</strong> 4-byte values), and if the starting position <strong>of</strong> allarrays is aligned on s bytes, then one cell <strong>of</strong> memory can be thought <strong>of</strong> as having s bytes.In this work, I restrict myself to the case where one cell <strong>of</strong> memory is equated to the size <strong>of</strong>one element <strong>of</strong> the array.For array-access analysis, representing and reasoning about arbitrary memory access patternsis not feasible. Thus array-access analysis is typically restricted to representing oneparticular class <strong>of</strong> memory access patterns. Linear Memory Access Descriptors or LMADs,first introduced by Paek et al. [17], are one such representation capable <strong>of</strong> precisely representinga wide variety <strong>of</strong> memory access patterns.Definition 1. An LMAD L is a mapping from a d-dimensional space <strong>of</strong> tuples <strong>of</strong> loopcounters to the 1-dimensional space <strong>of</strong> memory locations <strong>of</strong> the <strong>for</strong>m:d∑L(i 1 , i 2 , .., i d ) = c 0 + f k (i k ) (5.1)k=1where ∀(1 ≤ k ≤ d), starting with k = 1 and going to k = d:0 ≤ i k < u k (5.2)u k = g k (i 1 , i 2 , .., i k−1 ) (5.3)i k , u k , c 0 , f k (i k ), g k (i 1 , i 2 , .., i k−1 )ɛZ (5.4)where L is the LMAD, Z is the set <strong>of</strong> integers, i k is the k-th loop counter, c 0 to c d areinteger constants and u k is the upper bound <strong>of</strong> the k-th loop counter. Functions f k and g kare arbitrary integer functions.34