C. Lengauer. Loop parallelization in the polytope model. In International Conference on Concurrency Theory, pages 398--416, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....architectures. But, in the programming point of view, the question is: how such lowlevel imperative languages can safely describe problems Obviously they seem not abstract enough for reasoning on programs. The theory Pei [13, 14] was born from a generalization of the so called polytope model [7] and equational notations [9, 3] It was introduced in order to address crucial questions of program design methodology and transformations [5] Equational notations associated with the polytope model have been of great interest in processor array synthesis or compiling techniques for static ....

....data in a data parallel programming model. Here is an example dealing with parallelizing nested loops. 4.2. Time space transformation Among classical program transformations, the rewriting of loop nests is a parallelization technique a compiler could apply (see the wide literature on this topic [4, 1, 7], etc. Let us consider a simple example. Example 11. Nested Loops. Assuming that a i;1 ; i 2 [1: N ] and a 1;j ; j 2 [1: N ] are input data, let us consider the following sequential loop nest: do i=2,n do j=2,n a(i,j) a(i 1,j) a(i,j 1) enddo enddo From a dependence analysis an ane ....

C. Lengauer. Loop parallelization in the polytope model. Parallel Processing Letters, 4(3), 1994.

....a theory that can integrate di erent implementations like C or HPF. Then, we propose a language and its formal de nition in order to serve as a base for illustrating our theory. The theory that we propose is the theory Pei [17, 18] born from a generalization of the so called Polytope model [9]. Pei was introduced in order to address crucial questions of program design methodology and transformations [4] through a straightforward generalization of equational notations [11, 3] This paper is organized in three sections: section 2 gives an overview of the language, section 3 is devoted ....

C. Lengauer. Loop parallelization in the polytope model. Parallel Processing Letters, 4(3), 1994.

....between the loop nests or procedures. In order to allow these global transformations, the step has to be applied before any partitioning decision is taken w.r.t. the HW SW system being designed [12] To perform global loop transformations, we make use of a methodology based on the polytope model [19, 30]. In this model, each n level loop nest is represented geometrically by an n dimensional polytope. The order in which the iterations are executed can be represented by an ordering vector which traverses the polytope. To perform the transformations, we have developed a two phase approach. In the ....

C. Lengauer. Loop parallelization in the polytope model. In Proc. 4th Int. Conf. on Concurrency Theory (CONCUR), Hildesheim, Germany, Aug. 1993.

....Many theories have been proposed in order to derive and map programs onto target parallel architectures. Some of them (Alpha [Mau89] Crystal [CCL91] Lacs [Raj93] etc. define a program as a set of recurrence equations and propose synthesis techniques, which are founded on the Polytope Model [Len94]. Other ones (Linda [Gel85] Unity [CM88] Gamma [BM90] etc. are based on a refinement calculus : specifications are then expressed as predicates. The theory Pei was defined [VP92, VP93, Vio94] in order to unify these two approaches and benefit from their advantages. Pei provides a formal frame ....

C. Lengauer. Loop parallelization in the polytope model. to appear in PPL, 1994.

....of systems of parameterized ane recurrence equations which abstract single assignment loop nests. Using the dependence modeling in terms of utilization and emission sets [11] classical ane per variable space time transformations are computed [12] In this approach, called the polytope model [13], the information can be represented by parameterized convex polyhedra: utilization and emission sets, isotemporal spaces (i.e. polyhedra containing all the computation points scheduled at the same time step) etc. One of the techniques used to extract information from such polyhedra relies on ....

C. Lengauer. Loop parallelization in the polytope model. In CONCUR 93, 1993.

....Our paper focuses on this question. It discusses strategies of communication optimization for systems of parameterized ane recurrence equations (PARE s) 10] which formalize single assignment ane loop nests. The parallelization of a PARE is classically based on an ane space time transformation [5]. It assigns each computation to a virtual processor de ned by an allocation function and to an execution time de ned by a schedule function. The communications between virtual processors result from the projection of the dependences of the PARE. We distinguish between two types of ....

....by splitting the problem in subdomains relatively to p. Its dimension is dimEmitE;Y = Rank(R) It is contained in an ane space of dimension Rank(R) whose basis vectors are E;Y;j with 1 j dimEmitE;Y . A parallel solution to a set of PARE is characterized by an ane spacetime transformation [5]. It is expressed, for each variable X , by a full rank n n transformation matrix TX = X X where X is a d n integer matrix representing the allocation on a virtual processor space of dimension d , and X is a d n integer matrix de ning the multi dimensional schedule: t X (z) X ....

Lengauer, Ch.: Loop parallelization in the polytope model. CONCUR'93, LNCS 715, 398-416, 1993.

....and (b) constructing relationships between particular computations and between polytopes. A complete program may be represented by hierarchies of polytopes. Geometric representation of programs has been established by advances in parallelizing compiler research, in particular, the Polytope Model [2, 3, 4]. In the Polytope Model, programs, expressed in some language are rstly translated to a geometric representation. Optimization may be performed in the geometric domain and lastly, a translation back to the language representation completes the process. The typical bene t of these geometric ....

C. Lengauer, \Loop parallelization in the polytope model," CONCUR'93, 1993.

....Researches in automatic parallelization are especially focused on regular programming structures i.e. nested loops. In this context, the community have proposed a variety of tools and transformation techniques for this purpose. Recently, a mathematically based model, called the polytope model [1, 2, 5], was designed, in order to de ne a general framework for automatic parallelization of nested loops. The parallelization in this model consists in three steps. First, modeling the loop nest, called source program, by a convex polytope de ning its index space. The latter is represented by a system ....

....by scanning the target polytope through a nest of target loops. Each space dimension becomes a parallel loop and each time dimension becomes a sequential loop. The formalism on which this model is based, i.e. linear algebra, limits it to regular transformations. In fact, in the basic model [5], only ane transformations (represented by a matrix) can be treated. Lately, the model was extended, rst, to deal with singular matrices [6] and second, to allow the use of ane by statement and piecewise ane transformations [3, 9] Dealing with this second extension, in the code generation step, ....

[Article contains additional citation context not shown here]

C. Lengauer, Loop parallelization in the polytope model, CONCUR 93, Lecture Notes in Computer Science, No. 715, pp. 398-416, Springer Verlag, 1993.

....the evaluation of g on R g . Therefore, some temporary storage is needed on domain: T g = N g n R g : 1) Here, T g = fi; j j 0 i P 0 j Mg. But not only do we want to cope with temporary storage, we also try and extract parallelism in the evaluation of data fields. The polytope model [6, 7], developed for the automatic parallelization of imperative programs, happens to suit our purpose very well. This model simultaneously expresses the parallel execution order [8] and the processor allocation using space time mappings. The time mapping, a.k.a. the schedule, gives logical execution ....

C. Lengauer. Loop parallelization in the polytope model. In E. Best, editor, CONCUR '93, LNCS 715, pages 398--416. Springer-Verlag, 1993.

....of tiling is to coalesce sets of operations. Technically, we restrict ourselves to the case that these sets, the tiles, are (multidimensional) parallelepipeds. The problem area addressed in this paper is tiling for automatic parallelization, especially loop parallelization in the polytope model [3, 8], which is based on space time mapping, and extensions thereof (the polyhedron model [5] The methods of the polyhedron model typically aim at maximal parallelism, not maximal performance, since the latter is too complex to be computed during compilation. Hence, the resulting ne grained ....

C. Lengauer. Loop parallelization in the polytope model. In E. Best, editor, CONCUR'93, LNCS 715, pages 398-416. Springer-Verlag, 1993.

....the number of dimensions of its vector argument. That is to say, poly1 applies on a vector of type V ect n ff and polyk on a vector of type V ect n 1 ( V ect n k ff) The skeleton polyn p f 1 f 2 v applies the function f 1 to the elements of v contained in the n dimensional polytope [Len93] described by a predicate p (a set of inequalities between aOEne expressions) and the function f 2 on the elements outside the polytope. For example, poly2 , which takes a matrix (vectors of vectors) as argument, can be dened in Haskell as follows: poly2 p f g v = array (0,n 1) i,array ....

C. Lengauer. Loop parallelization in the polytope model. In International Conference on Concurrency Theory, LNCS 715, pages 398416, Hildesheim, Germany, 1993.

....in this paper, an additional restriction is that the tiles must be parallelepipeds, i.e. tiles in d dimensional space can be generated by d families of parallel hyperplanes [17] The focus of this paper is on tiling for automatic parallelization, esp. loop parallelization in the polytope model [12, 19], which is based on space time mapping. The methods of this model typically search for optimal parallelism, not optimal performance, since the latter is mathematically too complex to be computed during compilation. Consequently, the granularity of the automatically derived parallelism is very ne ....

C. Lengauer. Loop parallelization in the polytope model. In E. Best, editor, CONCUR'93, LNCS 715, pages 398-416. Springer-Verlag, 1993. 14

....purpose is to help identify the proper code generation methods for a space time mapped nest. We illustrate the hierarchy and its use on a loop nest for computing the reflexive transitive closure of a graph. 1 Introduction Traditional methods of space time mapping apply to nests of for loops [15]. Given a for loop nest, an optimizing search can identify at compile time a space time mapping which is minimal according to some stated metric (like the number of execution steps, processors, communication links, etc. This is so because all information necessary for the search is static, i.e. ....

....give an example inner loop on j . Class 4: Affine Loops. The bounds of these loops are affine expressions in the indices of the outer loops and in the structure parameters (i.e. the parameters which define the problem size) Nests with only affine loops can be treated well by traditional methods [15], which are realized in a number of systems [2, 14, 22, 23] Inner loop: for j : 0 to i 5 do. Class 3: Convex Loops. If the loop, together with the loops enclosing it, enumerates a (discrete) convex set (the execution space) then there must be a loop nest which enumerates precisely the points ....

C. Lengauer. Loop parallelization in the polytope model. In E. Best, editor, CONCUR'93, Lecture Notes in Computer Science 715, pages 398--416. SpringerVerlag, 1993.

....the MPI implementation of this schema is complex and still under development, no experimental results are available yet. Our aim is to have control over the time and space consumption to make parallelization accessible for automatic optimization, similarly to the tradition of loop parallelization [17]. Note that, although the actual number of processors assigned to a particular task depends on run time parameters, the structure of the parallel execution is determined by the skeletons at compile time. If this is not required, 6 C.A. Herrmann C. Lengauer one might consider using an ....

C. Lengauer. Loop parallelization in the polytope model. In E. Best, editor, CONCUR '93, LNCS 715, pages 398-416. Springer-Verlag, 1993.

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C. Lengauer. Loop parallelization in the polytope model. In International Conference on Concurrency Theory, pages 398--416, 1993.

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Christian Lengauer. Loop Parallelization in the Polytope Model. In Eike Best, editor, CONCUR'93, Lecture Notes in Computer Science 715, pages 398--416. Springer-Verlag, 1993.

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Christian Lengauer. Loop Parallelization in the Polytope Model. In Eike Best, editor, CONCUR'93, Lecture Notes in Computer Science 715, pages 398--416. Springer-Verlag, 1993.

No context found.

C. Lengauer. Loop Parallelization in the Polytope Model. In E. Best, editor, CONCUR'93, Lecture Notes in Computer Science 715, pages 398--416. Springer-Verlag, 1993.

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C. Lengauer. Loop parallelization in the polytope model. In International Conference on Concurrency Theory, LNCS 715, pages 398--416, Hildesheim, August 1993.

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C.Lengauer. "Loop parallelization in the polytope model", Proc. of the Fourth Intnl. Conf. on Concurrency Theory (CONCUR93), Hildesheim, Germany, Aug. 1993.

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C. Lengauer. Loop parallelization in the polytope model. Parallel Processing Letters, 4(3), 1994.

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C. Lengauer. Loop parallelization in the polytope model. In CONCUR 93, 1993.

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C. Lengauer. Loop parallelization in the polytope model. In CONCUR 93, 1993.

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C. Lengauer. Loop parallelization in the polytope model. In CONCUR 93, 1993.

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C. Lengauer. Loop parallelization in the polytope model. In E. Best, editor, CONCUR'93, Lecture Notes in Computer Science 715, pages 398--416. Springer Verlag, 1993.

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