P. Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53 (or 23--52??), February 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....dynamic single assignment (DSA) form. Since array elements are then assigned a value only once, they become equivalent to the values they were assigned, allowing much easier transformations on the data flow. Currently only a subset of imperative programs can be automatically converted to DSA form[6, 9]. Fortunately many multimedia kernels belong to that subset. Hence DSA is already widely used in the context of parallelization [7] and systolic array for (int j = 0; j 100; j ) if (j i 105) a[j i] j] in[j i] S1 else a[j i] j] a[j i 5] j 3] S2 for (int i = 0; i ....

....is of the following form: # #c . 1) Here A is a non singular n n matrix, # is a vector containing the iterators of the surrounding loops and #c is any column vector of length n. Note that this is often the case in DSA programs, e.g. the ones that are produced by the DSA conversion in [6]. This is a well known set of programs, both in the area of parallelization as well as the area of hardware synthesis [16] since these kinds of programs can be modeled and operated on using well established mathematical methods. The example from Fig. 2 satisfies all of the conditions above. ....

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P. Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53, 1991.

....is due to data transfer and storage. The DESICS group at IMEC has developed a Data Transfer and Storage Exploration (DTSE) methodology[4] that aims to reduce the memory requirements and data transfer costs of such designs. In a first step, the program is brought into a single assignment form[9] where each memory cell is written at most once. Typically, this further increases the memory usage of the program (as extra dimensions are introduced for arrays with elements that are written several times) However, this form simplifies the dependencies between read and write statements, and ....

....are not single assignment. However, spurious live areas will be introduced and one can expect a serious deteriorisation of the precision of the results. Another technique, also useful within the overall DTSE methodology, would be to develop tools to transform programs into single assignment form [9]. The other requirement is that bounds of for loops and conditions in if then else are linear in the iterators of the surrounding for loops. In absence of this, one could consider the use of known approaches within abstract 0 1 2 3 4 5 6 0 1 Fig. 4. Overlapping elements on boundary ....

P. Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53, 1991.

....assume that all processes run on a single CPU and each process is run to completion. Finally, it is assumed that when all the processes in the PDG are executed, the execution jumps to the first process and the entire execution sequence is repeated. Our focus in this paper is on affine programs [5] as defined by Feautrier. Data structures in these programs are restricted to be arrays and scalar variables, and control structures are limited to sequencing and nested loops. Each iteration of a given nested loop is represented by an iteration vector, which contains the values of the loop ....

P. Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--51, 1991.

....and su#cient equivalence conditions on the corresponding models of initial and transformed programs. Figure 2 puts our scheme in a nutshell. The technique we present in this paper is applicable for loop and data reuse transformations on sequential programs in dynamic single assignment form [9]. Outline of the paper. Section 2 discusses related work and contrasts the work presented in this paper with respect to it. Section 3 explains in brief the geometric model that we use in our verification technique. Section 4 describes the transformations that are targeted in this work. Section 5 ....

....3 Geometrical Program Modeling The novelty of our approach lies in the geometric model on which our technique is based. Geometrical models have in the past been quite extensively used to model and analyse execution of program statements, in the parallel compiler and regular array synthesis domain [9,17,18]. Though the model itself is quite simple, it concisely represents most of the necessary information about the data and control flow in the program. We use formulas that encode a#ne constraints on integer variables, logical connectives and quantifiers, also called Presburger formulas, to ....

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Feautrier, P., Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23-53. 1991.

....The ADG and the EDG have the same vertices, but the ADG has more edges, defined by: S i ; I) S j ; J) in the ADG) 9 e = S i ; S j ) in the RDG ) such that (I ; J) 2 D e : In some cases, output and anti dependences can be removed by data renaming and or expansion. See for example [Fea91] 6 For a certain class of nested loops, it is possible to express exactly this set of pairs (I ; J) see [Fea91] I is given as an affine function f i;j of J where J varies in a polyhedron P i;j : f(I ; J) 2 D j S i (I) S j (J)g = f(f i;j (J) J) j J 2 P i;j ae Dg (1) In most ....

....the ADG) 9 e = S i ; S j ) in the RDG ) such that (I ; J) 2 D e : In some cases, output and anti dependences can be removed by data renaming and or expansion. See for example [Fea91] 6 For a certain class of nested loops, it is possible to express exactly this set of pairs (I ; J) see [Fea91] I is given as an affine function f i;j of J where J varies in a polyhedron P i;j : f(I ; J) 2 D j S i (I) S j (J)g = f(f i;j (J) J) j J 2 P i;j ae Dg (1) In most dependence analysis algorithms however, rather than the set of pairs (I ; J) one computes the set E i;j of all possible ....

Paul Feautrier. Dataflow analysis of array and scalar references. Int. J. Parallel Programming, 20(1):23--51, 1991.

.... dependence level [AK82, AK87] dependence direction vector [Wol82, Wol89] dependence polyhedron cone [IT87] and more and more accurate tests for dependence analysis have been designed (Banerjee s tests [Ban88] I test [KKP90, PKK91] Delta test [GKT91] test [LYZ89, Gru90] PIP test [Fea91] PIPS test [IJT91] Omega test [Pug92] In general, dependence abstractions and dependence tests have been introduced with some particular loop transformations in mind. For example, the dependence level was designed for Allen and Kennedy s algorithm, whereas the PIP test is the main ....

....: In general, dependence abstractions and dependence tests have been introduced with some particular loop transformations in mind. For example, the dependence level was designed for Allen and Kennedy s algorithm, whereas the PIP test is the main tool for Feautrier s method for array expansion [Fea91] and parallelism extraction by affine schedulings [Fea92a, Fea92b] However, very few authors have studied, in a general manner, the links between both theories, dependence analysis and loop restructurations, and have tried to answer the following two dual questions: ffl What is the minimal ....

Paul Feautrier. Dataflow analysis of array and scalar references. Int. J. Parallel Programming, 20(1):23--51, 1991.

....allows us to perform all the optimizations together and enables us to decide the tradeoff between competing optimizations. Data flow analysis has been successfully applied to allow code optimizations involving scalar variables. Only recently has the technique been applied to ar ray variables. In [2], the author proposed a data flow analysis algorithm to collect information about array and scalar references and suggested that this information could be used for program verification and parallel program construction. We use data flow analysis for communication optimizations for which we need to ....

P. Feautrier, "Dataflow Analysis of Array and Scalar References," International Journal of Parallel Programming, Vol. 20, January, 1991.

....code to enumerate the data elements that are referenced by a set of nested loops. The same methods can be applied to loop transformations. Other work by Irigoin and Triolet [43] uses similar techniques for coarser grained parallelism. Others, independently, have used similar methods. Feautrier [24, 25, 26] has proposed an algorithm for generating loop bounds (the central problem for code generation) created by unimodular transformations on source programs. He uses a variant on the simplex algorithm which permits integer solutions and is fast on the sort of small problems arising in loop ....

P. Feautrier. Dataflow analysis of array and scalar references. Int. Journal of Parallel Programming, 20(1):23--53, Feb. 1991.

....exist on the code that can be handled in the present version however, some of which will be alleviated through future work. The main requirements are that the code is single assignment and has affine array indexes. This is achievable by a good array data flow analysis preprocessing, see [6] and [13] The single assignment format enables the data dependency analysis needed for generation of the underlying polyhedral dependency graph (PDG) model, 4] It also opens for code optimizations that can be guided by the storage requirement estimates. The methodology furthermore requires that ....

P. Feautrier, "Dataflow analysis of array and scalar references", International Journal of Parallel Programming, Vol. 20, No. 1, Feb. 1991, pp. 23-52

....text. NRS is the number of loops surrounding both R and S. One has x p y j x[1: p] y[1: p] x[p 1] y[p 1] and is given by x y j j xj Gamma1 x p y (1) The fact that operation hR; xi is executed before the operation hS; yi is written: hR; xi OE hS; yi. It is shown in [6] that: hR; xi OE hS; yi j x y ( x[1: N RS ] y[1: N RS ] R S) j NRS hR; xi OE p hS; yi (2) hR; xi OE p hS; yi , ae 0 p NRS : x p y p = NRS : x[1: N RS ] y[1: N RS ] R S (3) In our running example, we have: hS1; 2i OE 0 hS2; 3; 1i and hS2; 2; 3i OE 0 hS2; 3; 1i 2.2 ....

....flow dependences are artificial dependences too and are called spurious flow dependences. The removal of artificial dependences by data restructuring, is called data expansion. This technique will be detailed in the next section. Direct flow dependences are computed by data flow analysis [6]. It must determine for each memory cell c read by an operation w, the last operation in OE which gives a value to c before the execution of w. This operation is called the source of the read: source(c; w) max OE fv j v ffi wg (4) The computation of the source can be done by PIP (Parametric ....

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P. Feautrier. Dataflow Analysis of Array and Scalar References. Int. J. of Parallel Programming, 20(1):23-53, February 1991.

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P. Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53 (or 23--52??), February 1991.

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P. Feautrier, "Dataflow analysis of array and scalar references," Int. J. Parallel Programming, vol. 20, no. 1, pp. 23--53, Feb. 1991.

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P. Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53, February 1991.

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FEAUTRIER, P. 1991. Dataflow analysis of array and scalar references. Int. J. Parallel Program. 20, 1, 23--53.

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Paul Feautrier. Dataflow analysis of array and scalar references. Int. J. Parallel Programming, 20(1):23--51, 1991.

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Paul Feautrier. Dataflow analysis of array and scalar references. Int. J. Parallel Programming, 20(1):23--51, 1991.

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P. Feautrier. Dataflow analysis of array and scalar references. Int. J. of Parallel Programming, 20(1):23--51, 1991.

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Feautrier P., "Data flow Analysis of Array and Scalar References", Int. Journal of Parallel Prog., 20 (1), 1991.

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Paul Feautrier. Dataflow analysis of array and scalar references. Int. J. Parallel Programming, 20(1):23--51, 1991.

No context found.

Paul Feautrier. Dataflow analysis of array and scalar references. Int. J. Parallel Programming, 20(1):23--51, 1991.

No context found.

Paul Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53, January 1991.

No context found.

P. Feautrier. Dataflow analysis of array and scalar references. Int. J. of Parallel Programming, 20(1):23--51, 1991.

No context found.

P. Feautrier, "Dataflow analysis of array and scalar references", International Journal of Parallel Programming, Vol. 20, No. 1, Feb. 1991, pp. 23-52

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Paul Feautrier. Dataflow analysis of array and scalar references. International Journal of Parallel Programming, 20(1):23--53, January 1991.

No context found.

P. Feautrier. Dataflow analysis of array and scalar references. Int. J. of Parallel Programming, 20(1):23--51, 1991.

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