Array Dataflow Analysis

We present a unified mathematical framework for analyzing the tradeoffs between parallelism and storage allocation within a parallelizing compiler. Using this framework, we show how to find a good storage mapping for a given schedule, a good schedule for a given storage mapping, and a good storage mapping that is valid for all legal (one-dimensional affine) schedules. We consider storage mappings that collapse one dimension of a multidimensional array, and programs that are in a single assignment form and accept a one-dimensional affine schedule. Our method combines affine scheduling techniques with occupancy vector analysis and incorporates general affine dependences across statements and loop nests. We formulate the constraints imposed by the data dependences and storage mappings as a set of linear inequalities, and apply numerical programming techniques to solve for the shortest occupancy vector. We consider our method to be a first step towards automating a procedure that finds the optimal tradeoff between parallelism and storage space.

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“…The dependences P j are determined using an array dataflow analysis [Feautrier 1991[Feautrier , 2001aMaydan et al 1993;Pugh 1992].…”

Section: A Step Towards Unifying Schedule and Storage Optimizationmentioning

confidence: 99%

A step towards unifying schedule and storage optimization

Thies

Vivien

Amarasinghe

2007

ACM Trans. Program. Lang. Syst.

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“…under constraints (6) and (7). We define this problem (the objective function (8) and the constraints (6) and 7) as Problem 0 and the problem in Theorem 1 as Problem 1.…”

Section: Appendix A: Proof Of Lemmamentioning

confidence: 99%

“…Therefore, the optimal objective function value of (8) is equal to or greater than the optimal objective function value of (5). Similarly, for any optimal solution for Problem 1 over G , based on Lemmas 7 and 8, we can find an optimal solution whose objective function value for (5) is equal to that for (8). Therefore, the optimal objective function value for (5) is equal to or greater than the optimal objective function value of (8…”

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confidence: 96%

“…An input array region is upwardly exposed to the beginning of L 1 .An output array region is live after L m . A local array region does not intersect with any input or output array regions.By utilizing the existing dependence analysis, region analysis and live analysis techniques[5,8,10,13,15], we can compute input, output and local array regions efficiently.Note that input and output regions may overlap with each other. In the Jacobi example(Figure 1), temp(2:N-1,2:N-1) is the local array region, A(1:N,1:N) is the input region, and A(2:N-1,2:N-1) is the output region.…”

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confidence: 99%

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Improving data locality by array contraction

Song

Wang

et al. 2004

IEEE Trans. Comput.

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Array contraction is a program transformation which reduces array size while preserving the correct output. In this paper, we present an aggressive array-contraction technique and study its impact on memory system performance. This technique, called controlled SFC, combines loop shifting and controlled loop fusion to maximize opportunities for array contraction within a given loop nesting. A controlled fusion scheme is used to prevent over-fusing loops and to avoid excessive pressure on the cache and the registers. Reducing the array size increases data reuse because of the increased average number of memory operations on the same memory addresses. Furthermore, if the data size of a loop nest fits in the cache after array contraction, then repeated references to the same variable in the loop nest will generate cache hits, assuming set conflicts are eliminated successfully.

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“…I also formalized a comparison algorithm, which is needed when there are several potential sources. But it was not until [4] that I managed to prove its termination.…”

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confidence: 93%

Author retrospective for array expansion, array shrinking, or there and back again

Feautrier

2014

25th Anniversary International Conference on Supercomputing Anniversary Volume -

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In the late 1980's, I was recovering from a long stint as the manager of Paris University computing facility (yes, there were still computing facilities, and PCs were just coming of age) and I was teaching old fashioned automatic parallelization to postgraduate students. I had heard of scalar expansion and privatization, and it was a natural question whether it could be extended to arrays. I first concocted partial solutions -for instance, valid only for the innermost loop -and then decided to try for the general case: converting array accesses to single assignment form. I soon realized that this implied finding the source, or last write before a given read, and that the solution must be a function of the position of the read in the temporal execution of the program. It was obvious that this could not be done for arbitrary complex programs, hence I specified a set of restriction: the polyhedral model. I also introduced the execution order, now known as the "happens-before" relation. Finding the last write then became an integer programming problem with some unfamiliar features: lexicographic order took the place of the economic function, the problem had to be solved exactly, and the coordinates of the read operation were acting as parameters. Hence,

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Array Dataflow Analysis

Cited by 4 publications

References 22 publications

A step towards unifying schedule and storage optimization

A step towards unifying schedule and storage optimization

Improving data locality by array contraction

Author retrospective for array expansion, array shrinking, or there and back again

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