P. Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I, one dimensional time, and part II, multidimensional time. International Journal of Parallel Programming, vol. 21, n 5-6, octobre 1992, pp. 313348, 389420.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....abstraction it uses is sharper than the one in Allen and Kennedy s algorithm. However, the structure of the RDG is not considered. It is optimal for maximal parallelism detection if the only information on direction vectors with no knowledge of the dependence graph structure. Feautrier s algorithm [19, 20] produces a general aOEne transformation. It can handle perfectly nested loops as well as non perfectly nested loops as long as exact dependence analysis is feasible. It relies on aOEne dependences. The aOEne transformation is build as a solution of linear programs obtained by the aOEne form of ....

.... Distribution Optimal Yes Very easy No Non perfect Direction vectors Wolf Lam [36] One statement Unimodular Optimal No Easy Yes Perfect Polyhedra Darte Vivien [14] statements Shifted linear Optimal Partial yes Quite easy Yes Perfect AOEne (exact) Feautrier [20] statements AOEne Sub optimal No Complicated No Non perfect AOEne (exact) Lim Lam [30] statements AOEne Sub optimal Yes Non perfect Table 1: A comparison of various loop parallelizing algorithms. 10 system were each loop index depends only on the previous loop ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part II: multidimensional time. Int. J. Parallel Programming, 21(6):389420, December 1992.

....abstraction it uses is sharper than the one in Allen and Kennedy s algorithm. However, the structure of the RDG is not considered. It is optimal for maximal parallelism detection if the only information on direction vectors with no knowledge of the dependence graph structure. Feautrier s algorithm [19, 20] produces a general aOEne transformation. It can handle perfectly nested loops as well as non perfectly nested loops as long as exact dependence analysis is feasible. It relies on aOEne dependences. The aOEne transformation is build as a solution of linear programs obtained by the aOEne form of ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I: onedimensional time. Int. J. Parallel Programming, 21(5):313348, October 1992.

....loop transformations, and in the systolic array community for the automatic synthesis of systolic arrays. In 1988, Paul Feautrier introduced PIP [12] a software tool for parametric (integer) linear programming, and he demonstrated its interest for dependence analysis [13] loop parallelization [14, 15], code generation [6] among others. This work initiated further developments based on polytopes. The aim of this paper is to give an introduction to these techniques, mainly by showing how they have been progressively used, over the last past 25 years, for studying repetitive computations: rst in ....

....latency of particular regular schedules, called shifted linear multi dimensional schedules. Such a recursive decomposition is also used in most loop parallelization algorithms, for example in Allen and Kennedy s algorithm [1] in Darte and Vivien s algorithm [11] and in Feautrier s algorithm [15] (see Section 4) For each of them, a similar subgraph G can be dened, but its denition depends on the representation of dependences that is used. Consider the system of uniform recurrence equations of Example 1 and whose dependence graph is depicted on Figure 2. The two edges from a 1 to ....

[Article contains additional citation context not shown here]

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part II: multidimensional time. Int. J. Parallel Programming, 21(6):389420, December 1992.

....loop transformations, and in the systolic array community for the automatic synthesis of systolic arrays. In 1988, Paul Feautrier introduced PIP [12] a software tool for parametric (integer) linear programming, and he demonstrated its interest for dependence analysis [13] loop parallelization [14, 15], code generation [6] among others. This work initiated further developments based on polytopes. The aim of this paper is to give an introduction to these techniques, mainly by showing how they have been progressively used, over the last past 25 years, for studying repetitive computations: rst in ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I: onedimensional time. Int. J. Parallel Programming, 21(5):313348, October 1992.

.... [2, 3, 4, 5] and synthesis of regular arrays [6, 7, 8, 9] Works on the formalism of recurrence equations lead to various languages like Lucid [10] Lustre [11] Signal [12] Crystal [13] Pei [14] and Alpha [15] Works on the scheduling of recurrence equations lead to important results [16, 17, 8, 3, 18]. In this paper, we study systems of recurrence equations (SAREs) withx the formalism of the Alpha language. Recently, Alpha was extended to include program structuring [19] This allowed to write and manipulate big SAREs speci cations in a structured form, but raised several questions. For ....

....used in systolic synthesis. The structuring of Alpha has already been presented in [19, 20] but its understanding is fundamental for the rest of the paper. In section 4, we explain how to build an eOEcient schedule tool for SAREs. As the foundations of the scheduling method has been presented [3, 18, 17, 9], this part consists in results from practical scheduling and high level built in options that should provide a schedule tool for vlsi synthesis. Section 5 deals with structured linear scheduling and isolates some cases of structured SAREs where a structured linear scheduling can be found. Then a ....

[Article contains additional citation context not shown here]

P. Feautrier. Some eOEcient solution to the aOEne scheduling problem, part II, multidimensional time. Int. J. of Parallel Programming, 21(6), December 1992.

.... 34 8 Conclusion 39 RR n3282 4 Tanguy Risset, Florent Dupont de Dinechin, Sophie Robert 1 Introduction Recurrence equations where introduced as a formalism for describing computation by Karp, Miller and Winograd [1] This formalism has been widely studied in the eld of parallelization [2, 3, 4, 5] and synthesis of regular arrays [6, 7, 8, 9] Works on the formalism of recurrence equations lead to various languages like Lucid [10] Lustre [11] Signal [12] Crystal [13] Pei [14] and Alpha [15] Works on the scheduling of recurrence equations lead to important results [16, 17, 8, 3, 18] In ....

.... [2, 3, 4, 5] and synthesis of regular arrays [6, 7, 8, 9] Works on the formalism of recurrence equations lead to various languages like Lucid [10] Lustre [11] Signal [12] Crystal [13] Pei [14] and Alpha [15] Works on the scheduling of recurrence equations lead to important results [16, 17, 8, 3, 18]. In this paper, we study systems of recurrence equations (SAREs) withx the formalism of the Alpha language. Recently, Alpha was extended to include program structuring [19] This allowed to write and manipulate big SAREs speci cations in a structured form, but raised several questions. For ....

[Article contains additional citation context not shown here]

P. Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I, one dimensional time. Int. J. of Parallel Programming, 21(5):313348, October 1992.

....abstraction it uses is sharper than the one in Allen and Kennedy s algorithm. However, the structure of the RDG is not considered. It is optimal for maximal parallelism detection if the only information on direction vectors with no knowledge of the dependence graph structure. Feautrier s algorithm [19, 20] produces a general aOEne transformation. It can handle perfectly nested loops as well as non perfectly nested loops as long as exact dependence analysis is feasible. It relies on aOEne dependences. The aOEne transformation is build as a solution of linear programs obtained by the aOEne form of ....

.... Optimal Yes Very easy No Non perfect Direction vectors Wolf Lam [36] One statement Unimodular Optimal No Easy Yes Perfect Polyhedra Darte Vivien [14] Multiple statements Shifted linear Optimal Partial yes Quite easy Yes Perfect AOEne (exact) Feautrier [20] Multiple statements AOEne Sub optimal No Complicated No Non perfect AOEne (exact) Lim Lam [30] Multiple statements AOEne Sub optimal Yes Non perfect Table 1: A comparison of various loop parallelizing algorithms. 10 system were each loop index depends only on ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part II: multidimensional time. Int. J. Parallel Programming, 21(6):389420, December 1992.

....abstraction it uses is sharper than the one in Allen and Kennedy s algorithm. However, the structure of the RDG is not considered. It is optimal for maximal parallelism detection if the only information on direction vectors with no knowledge of the dependence graph structure. Feautrier s algorithm [19, 20] produces a general aOEne transformation. It can handle perfectly nested loops as well as non perfectly nested loops as long as exact dependence analysis is feasible. It relies on aOEne dependences. The aOEne transformation is build as a solution of linear programs obtained by the aOEne form of ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I: onedimensional time. Int. J. Parallel Programming, 21(5):313348, October 1992.

....the bunch of known parallelism detection and code optimization algorithms, and those implemented in real compilers. Indeed, these algorithms are often diOEcult to implement, because they use graph manipulations, linear algebra, linear programming, and complex code restructuring (see for example [11, 23, 15, 9] for some parallelism detection algorithms) Consequently, their implementation is a research problem by itself and must be ease by a simple but powerful representation of the input program. This representation should provide all the basic blocks to let the researcher concentrate on algorithmic ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part II: multidimensional time. Int. J. Parallel Programming, 21(6):389420, December 1992.

....loop transformations, and in the systolic array community for the automatic synthesis of systolic arrays. In 1988, Paul Feautrier introduced PIP [12] a software tool for parametric (integer) linear programming, and he demonstrated its interest for dependence analysis [13] loop parallelization [14, 15], code generation [6] among others. This work initiated further developments based on polytopes. The aim of this paper is to give an introduction to these techniques, mainly by showing how they have been progressively used, over the last past 25 years, for studying repetitive computations: rst in ....

....latency of particular regular schedules, called shifted linear multi dimensional schedules. Such a recursive decomposition is also used in most loop parallelization algorithms, for example in Allen and Kennedy s algorithm [1] in Darte and Vivien s algorithm [11] and in Feautrier s algorithm [15] (see Section 4) For each of them, a similar subgraph G 0 can be dened, but its denition depends on the representation of dependences that is used. Back to Example 1 Consider the system of uniform recurrence equations of Example 1 and whose dependence graph is depicted on Figure 2. The two ....

[Article contains additional citation context not shown here]

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part II: multidimensional time. Int. J. Parallel Programming, 21(6):389420, December 1992.

....loop transformations, and in the systolic array community for the automatic synthesis of systolic arrays. In 1988, Paul Feautrier introduced PIP [12] a software tool for parametric (integer) linear programming, and he demonstrated its interest for dependence analysis [13] loop parallelization [14, 15], code generation [6] among others. This work initiated further developments based on polytopes. The aim of this paper is to give an introduction to these techniques, mainly by showing how they have been progressively used, over the last past 25 years, for studying repetitive computations: rst in ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I: onedimensional time. Int. J. Parallel Programming, 21(5):313348, October 1992.

....dependence abstraction it uses is sharper than the one in Allen and Kennedy s algorithm. However, the structure of the RDG is not considered. It is optimal for maximal parallelism detection if dependences are captured by direction vectors (and no dependence graph structure) Feautrier s algorithm [20, 21] produces a general aOEne transformation. It can handle perfectly nested loops as well as non perfectly nested loops as long as exact dependence analysis is feasible. It relies on aOEne dependences. The aOEne transformation is built as a solution of linear programs obtained by the aOEne form of ....

.... [2] Multiple statements Distribution Optimal Yes Very easy No Non perfect Direction vectors Wolf Lam [37] One statement Unimodular Optimal No Easy Yes Perfect Polyhedra Darte Vivien [15] Multiple statements Shifted linear Optimal Partial yes Quite easy Yes Perfect AOEne (exact) Feautrier [21] Multiple statements AOEne Sub optimal No Complicated No Non perfect AOEne (exact) Lim Lam [31] Multiple statements AOEne Sub optimal Yes Non perfect Table 1: A comparison of various loop parallelizing algorithms. depends only on the previous loop indices and on parameters. As many ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part II: multidimensional time. Int. J. Parallel Programming, 21(6):389420, December 1992.

....dependence abstraction it uses is sharper than the one in Allen and Kennedy s algorithm. However, the structure of the RDG is not considered. It is optimal for maximal parallelism detection if dependences are captured by direction vectors (and no dependence graph structure) Feautrier s algorithm [20, 21] produces a general aOEne transformation. It can handle perfectly nested loops as well as non perfectly nested loops as long as exact dependence analysis is feasible. It relies on aOEne dependences. The aOEne transformation is built as a solution of linear programs obtained by the aOEne form of ....

Paul Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I: onedimensional time. Int. J. Parallel Programming, 21(5):313348, October 1992.

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P. Feautrier. Some eOEcient solutions to the aOEne scheduling problem, part I, one dimensional time, and part II, multidimensional time. International Journal of Parallel Programming, vol. 21, n 5-6, octobre 1992, pp. 313348, 389420.

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