Paul Feautrier. Semantical analysis and mathematical programming; application to parallelization and vectorization. In M. Cosnard, Y. Robert, P. Quinton, and M. Raynal, editors, Workshop on Parallel and Distributed Algorithms, Bonas, pages 309--320. North Holland, 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....est le graphe de d ependances. Il en existe plusieurs variantes, dont le data ow graph, le 8 data dependence graph [71] 5] le program dependence graph [45] ou le dependence ow graph [84] L utilisation de ces graphes dans le cas particulier de la parall elisation est d ecrit dans [95] [43], 5] ou [33] et pour la comparaison de sous programmes dans [86] ou [6] Consid erant l exemple centaur, on peut d eplorer le trop petit nombre d applications en vraie grandeur, c est a dire produisant des outils s appliquant a des programmes r eels, de mani ere ecace. On soulignera ....

Feautrier P. "Semantical analysis and mathematical programming; application to parallelization and vectorization". In Workshop on Parallel and Distributed Algorithms, pages 309-320, Bonas, 1989. M. Cosnard, Y. Robert, P. Quinton, M. Raynal, editors, North Holland.

....a constrained way of describing a polyhedron since each loop bound can only depend on outer loop bounds; remember the special structure of A. Algorithms for converting such a set into a form that can be enumerated by loop structures can be found in Wolf and Lam [76] Irigoin [41] Feautrier [25], and Ancourt [4] Ribas [69] uses an algorithm that covers only simple cases (linear loop bounds) and decomposes the target space when necessary. However, not every point within the convex hull of T S necessarily corresponds to a point in I; when has rational components, it maps only a subset ....

....inefficiencies. The inefficiencies are due to redundant loop bounds, some of which may not be discarded during the simplification process. Consequently, researchers in both fields restrict themselves to unimodular transformations, for example, Ancourt [4] Banerjee [7] Dowling [22] Feautrier [25], Irigoin [41] Leverge, Mauras and Quinton [52] Ribas [69] and Wolf and Lam [76] As it turns out, the basic transformations in parallelizing compilation correspond to elementary matrices that are unimodular [7] Moreover, the composition of the transformations corresponds to the matrix ....

[Article contains additional citation context not shown here]

P. Feautrier. Semantical analysis and mathematical programming. In M. Cosnard, Y. Robert, P. Quinton, and M. Raynal, editors, Parallel and Distributed Algorithms, pages 309--320. North-Holland, 1989.

....These relations classically form a graph, known as the dependence graph. There exist may variants, such as the data AEow graph, the data dependence graph [71] 5] the program dependence graph [45] or the dependence AEow graph [84] Use of these graphs for parallelization is described in [95] [43], 5] or [33] and for comparison of program fragments in [86] or [6] Let us now focus on the state of the art for our two short term goals, automatic dioeerentiation and parallelization. 2.2.2 Automatic Dioeerentiation Automatic Dioeerentiation (A.D. is a technique that, given a program P ....

Feautrier P. "Semantical analysis and mathematical programming; application to parallelization and vectorization". In Workshop on Parallel and Distributed Algorithms, pages 309320, Bonas, 1989. M. Cosnard, Y. Robert, P. Quinton, M. Raynal, editors, North Holland.

....loop index p are given with respect to the inner loop index t . We must transform the polytope (AT Gamma1 ; b) to an equivalent one, A 0 ; b 0 ) whose defining inequations refer only to the indices of enclosing loops. Several algorithms for calculating A 0 and b 0 have been proposed [1, 16, 44, 54]; most of them are based on Fourier Motzkin elimination [47] a technique by which one eliminates variables from the inequalities. Geometrically, this projects the polytope on the different axes of the target coordinate system to obtain the bounds in each dimension. To bound the target polytope ....

....design [51] The application domain for which the system lends most support at present covers models and methods for the design of regular architectures. In this domain, the polytope model is used extensively [19] PAF. This is the automatic FORTRAN parallelizer of Paul (A. Feautrier [16]. The system converts nested DO loops to single assignment form [17] and uses parametric integer programming [15] to find a time minimal, not necessarily affine, shared memory parallel schedule. The source loops need not be perfectly nested. Presage. The novelty of Presage [53] was that it dealt ....

P. Feautrier. Semantical analysis and mathematical programming. In M. Cosnard, Y. Robert, P. Quinton, and M. Raynal, editors, Parallel & Distributed Algorithms, pages 309--320. North-Holland, 1989.

....a constrained way of describing a polyhedron since each loop bound can only depend on outer loop bounds; remember the special structure of A. Algorithms for converting such a set into a form that can be enumerated by loop structures can be found in Wolf and Lam [76] Irigoin [41] Feautrier [25], and Ancourt [4] Ribas [69] uses an algorithm that covers only simple cases (linear loop bounds) and decomposes the target space when necessary. However, not every point within the convex hull of T S necessarily corresponds to a point in I; when T Gamma1 has rational components, it maps only ....

....inefficiencies. The inefficiencies are due to redundant loop bounds, some of which may not be discarded during the simplification process. Consequently, researchers in both fields restrict themselves to unimodular transformations, for example, Ancourt [4] Banerjee [7] Dowling [22] Feautrier [25], Irigoin [41] Leverge, Mauras and Quinton [52] Ribas [69] and Wolf and Lam [76] As it turns out, the basic transformations in parallelizing compilation correspond to elementary matrices that are unimodular [7] Moreover, the composition of the transformations corresponds to the matrix product ....

[Article contains additional citation context not shown here]

P. Feautrier. Semantical analysis and mathematical programming. In M. Cosnard, Y. Robert, P. Quinton, and M. Raynal, editors, Parallel and Distributed Algorithms, pages 309--320. North-Holland, 1989.

....when one use speci cation languages like Alpha) or when the loop nest is submitted to a unimodular transform. This situation is characterized by the fact that all integer points in the given polyhedron are to be visited. In this form, the problem was rst solved by Irigoin in [Iri87] see also [Fea89, AI91, CFR95]. However, not all parallelizing transformations are unimodular. They may even be singular: this situation occurs when constructing communication loops [PR95] The solution is to write the transformation T = HU where U is unimodular and H is a Hermite normal form of T . One rst scans the image of ....

Paul Feautrier. Semantical analysis and mathematical programming. In M. Cosnard and al., editors, Parallel and distributed algorithms, pages 309320, North Holland, 1989. Elsevier Science Publishers B.V.

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Paul Feautrier. Semantical analysis and mathematical programming; application to parallelization and vectorization. In M. Cosnard, Y. Robert, P. Quinton, and M. Raynal, editors, Workshop on Parallel and Distributed Algorithms, Bonas, pages 309--320. North Holland, 1989.

.... by such a transformation will look like: 1 To see why this time may be multidimensional, see [Fea92b] 4 DO t= lb t (m; n) ub t (m; n) DOALL (x= lb x (m; n; t) ub x (m; n; t) DOALL (y= lb y (m; n; t; x) ub y (m; n; t; x) S ENDDOALL ENDDOALL ENDDO Methods of dependence analysis [Fea89, Fea91] and schedule computation [Fea92a, Fea92b, QR89, Qui87] are now well understood and provide an appropriate matrix U . In the transformation examples above, the parallelizer has to compute new loop bounds, and this is all but obvious. The rest of this paper concentrates on the algorithm for this ....

....be maxima minima of R1 or R2 expressions. Thus, the output is a list of bounds: first, the bounds of the outermost counter x 1 , then the bounds on x 2 as a function of the structure parameters plus x 1 , and so on for all entries in x. 4. 3 The Algorithm The basic method has been presented in [Fea89]. The first question is to know how one can exactly scan the integer points of D( z) that is, to build the L loop nest. The method proceeds by constructing n polyhedra: D k (x 1 : x k ; z) f(x k 1 : x n ) T j C x C 0 z b 0g for k = 0; n Gamma 1. Obviously, D( z) D 0 ....

P. Feautrier. Semantical analysis and mathematical programming; application to parallelization and vectorization. In P. Quinton M. Cosnard, Y. Robert and M. Raynal, editors, Workshop on Parallel and Distributed Algorithms, Bonas, pages 309--320. North Holland, 1989.

....contention. In fact, we will show in 3.4 how to model these effects as random fluctuations in the execution times. 2.1 The Dataflow Graph For a static control program, it is possible to analyze the flow of data through the operations and the memory cells. The basic technique is presented in [7] and in more details in [5] a brief description follow. For each read access in the program, the set of all preceeding write accesses to the same memory cell is characterized and its temporal maximum is computed. The result is the source of the value obtained by the read access. A source is ....

....f3g, F (3s Gamma 1) instances of f6g and F (3s) contains both instances of f2g and f5g. This suggests rewriting the original program in the form: DO 1 s = 1, n F(3s 2) F(3s 1) F(3s) 1 CONTINUE The explicit coding of each front is now a straightforward problem of loop transformation, see [7] for a general solution. The result is: DO 1 s = 1,n p(s) 1.0 sqrt(x(s) DOALL 2 j = s 1, n 2 a(j,s) z(j,s) p(s) PCASE DOALL 3 i = s 1,n 3 x(i) x(i) a(i,s) 2 3 We will use the parallel programming primitives of [11] Preliminary Version, May 23, 1989 8 PAR DOALL 4 i = s 1, n ....

[Article contains additional citation context not shown here]

Paul Feautrier. Semantical analysis and mathematical programming; application to parallelization and vectorization. In Workshop on Parallel and Distributed Algorithms, Bonas, October 1988.

....in Tableau, and each vector exactly has n 1 p entries. ffl In a similar way, Context is a list of Vectors. Each Vector represents a row of Matrix M followed by the corresponding entry in vector h. Context thus includes m Vectors of p 1 entries. 2.1. 1 Example This example is taken from [2]. We consider the loop nest below: for i: 0 to m do for j : 0 to n do II for k : 0 to i j do . and we wish to rewrite this nest in the order k, j, i. The bounds on k can easily be guessed (0 k m n) so let s look for the lower bound on j in the rewritten nest. This lower bound on j ....

....the solution The solution of a parametric problem may be in the form of a quast all of whose leaves are nil. This means in fact that the original polyhedron is empty whatever the values of the parameters. An example, due to Dirk Fimmel, is the following: i j 1) m n) 2 2 7 0 1 1 (#[2 6 9 0 0] #[5 3 0 0 0] #[2 10 15 0 0] #[ 2 6 3 0 0] #[ 2 6 17 0 0] #[0 1 0 1 0] #[1 0 0 0 1] Without the z option, the solution is: i j 1) m n) 1 ) if #[ 4 0 5] if #[ 0 4 3] if #[ 0 2 9] if #[ 0 2 3] newparm 2 (div #[ 0 2 3] 6) newparm 3 (div #[ 0 2 10 7] 12) ....

[Article contains additional citation context not shown here]

Paul Feautrier. Semantical analysis and mathematical programming; application to parallelization and vectorization. In M. Cosnard, Y. Robert, P. Quinton, and M. Raynal, editors, Workshop on Parallel and Distributed Algorithms, Bonas, pages 309--320. North Holland, 1989.

....when one use specification languages like ALPHA) or when the loop nest is submitted to a unimodular transform. This situation is characterized by the fact that all integer points in the given polyhedron are to be visited. In this form, the problem was first solved by Irigoin in [9] see also [8, 1, 5]. However, not all parallelizing transformations are unimodular. They may even be singular: this situation occurs when constructing communication loops [11] The solution is to write the transformation T = HU where U is unimodular and H is a Hermite normal form of T . One first scans the image of ....

P. Feautrier. Semantical analysis and mathematical programming. In M. C. et al., editor, Parallel and distributed algorithms, pages 309--320, North Holland, 1989. Elsevier Science Publishers B.V.

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