## Automatic Parallelization in the Polytope Model (1996)

Venue: | Laboratoire PRiSM, Université des Versailles St-Quentin en Yvelines, 45, avenue des États-Unis, F-78035 Versailles Cedex |

Citations: | 56 - 3 self |

### BibTeX

@INPROCEEDINGS{Feautrier96automaticparallelization,

author = {Paul Feautrier},

title = {Automatic Parallelization in the Polytope Model},

booktitle = {Laboratoire PRiSM, Université des Versailles St-Quentin en Yvelines, 45, avenue des États-Unis, F-78035 Versailles Cedex},

year = {1996},

pages = {79--103},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The aim of this paper is to explain the importance of polytope and polyhedra in automatic parallelization. We show that the semantics of parallel programs is best described geometrically, as properties of sets of integral points in n-dimensional spaces, where n is related to the maximum nesting depth of DO loops. The needed properties translate nicely to properties of polyhedra, for which many algorithms have been designed for the needs of optimization and operation research. We show how these ideas apply to scheduling, placement and parallel code generation. R'esum'e Le but de cet article est d'expliquer le role jou'e par les poly`edres et les polytopes en parall'elisation automatique. On montre que la s'emantique d'un programme parall`ele se d'ecrit naturellement sous forme g'eom'etrique, les propri'et'es du programme 'etant formalis'ees comme des propri'et'es d'ensemble de points dans un espace `a n dimensions. n est li'e `a la profondeur maximale d'imbrication des boucles DO. Les...

### Citations

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Citation Context ...or modifications. A limited amount of knowledge was then added to improve the final result, for instance in the form of a type system. In the case of a fully developed type system, like the one in ML =-=[MTH90]-=-, knowledge about operators in the language is given to the compiler in the form of typing rules, which are essentially Horn clauses. A program is correct if, for each of its expressions, one can prov... |

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Citation Context ...next section the tools which are needed for such calculations. 3 Basic tools for handling polyhedra and Z-polyhedra The basic reference on linear inequalities in rationals or integers is the treatise =-=[Sch86]-=-. 3.1 Polyhedra and polytopes There are two ways of defining a polyhedron. The simplest one is to give a set of linear inequalities: Ax + as0: The polyhedron is the set of all x which satisfies these ... |

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Citation Context ... question of their emptiness or not. For canonical Z-polyhedra, this is the linear integer programming question [Sch86, Min83]. I will briefly sketch two integer programming algorithm: the Omega test =-=[Pug91a]-=- which is an extension of Fourier-Motzkin, and the Gomory cut method, which is an extension of the Simplex [Gom63]. Recall that in the Fourier-Motzkin method, we start by extracting lower and upper bo... |

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Citation Context ...esearchers use methods which find only parts of T . The problem is then to extend T to a one-to-one transformation, or to fit the parts together. Scheduling Since the pioneering papers of [KMW67] and =-=[Lam74], there ha-=-ve been a large number of papers on scheduling, mainly from the "systolic" community. The basic observation is that for any function ` from the set of operations to any totally ordered set, ... |

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Citation Context ...94]. Other researchers use methods which find only parts of T . The problem is then to extend T to a one-to-one transformation, or to fit the parts together. Scheduling Since the pioneering papers of =-=[KMW67] and [Lam7-=-4], there have been a large number of papers on scheduling, mainly from the "systolic" community. The basic observation is that for any function ` from the set of operations to any totally o... |

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Citation Context ...ension of x and y. -- N RS is the number of loops surrounding both R and S. Accordingly, the number of loop surrounding S should be written N SS . It will be abbreviated to N S here. We have shown in =-=[Fea91]-=- that: hR; xi OE hS; yi j x[1::N RS ]sb[1::N RS ]s(x[1::N RS ] = y[1::N RS ] R! S): (3) The predicate OE is not convex, hence it cannot be represented as a polyhedron. However, OE can be split into N ... |

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Citation Context ...dition for determinism: we have already sacrificed some parallelism for simplicity. Computing ffi may be of arbitrary complexity. However, a sufficient condition for commutation is easily constructed =-=[Ber66]: let R(u) (res-=-p. M (u)) be the set of memory cells which are read (resp modified) by u. u and v commute if: M (u) " R(v) = ;; R(u) " M (v) = ;; M (u) " M (v) = ;: The three terms in that formula appe... |

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Citation Context ...and g are the subscripting functions; they have the same number of components, namely the rank of array X. The operations in dependence at depth p are the members of the following dependence relation =-=[PW93]-=-: fhR; xi; hS; yi j Q p RS (x; y)g where Q p RS is the following polytope: Q p RS (x; y) j f (x) = g(y)shR; xi OE p hS; yisx 2 D Rsy 2 D S (4) The union of all dependence relations is a symbolic descr... |

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Citation Context ...complicated bounds than is necessary, unless one programs a redundancy eliminator. Another solution is to use PIP for computing maxima and minima, in which case redundancy is automatically eliminated =-=[CBF95]-=-. Non-unimodular transformations In case T is not unimodular, the solution is to build its Hermite normal form T = HU [Dar93, Xue94]. One builds, according to the above method, a loop nest which scans... |

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