Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from a given set. It is simpler to verify an inclusion condition. In this case we just have to verify that a condition is verified for one value from a given set. Hence an inclusion condition can be modelized by an integer programming problem and can be resolved by a software such as PIP (see [9]) So, in place of counting the number of elements in C(S) we compute the number of elements in the set C(S) W S (J) T (J) 2 T ( Phi R S (I) 5) The final result is obtained using the following relation: Card(C(S) Card( W S (J) Gamma Card C(S) 6) The PIP ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....compute (3) broadly fall in three classes: ffl The resolution of (2) using Presburger formula. This method has been pioneered by Pugh [32] ffl The resolution of (3) using integer linear programming. This method was initiated by Feautrier [17] who crafted for this purpose a software called PIP [16], based on the simplex, to directly compute max . ffl The execution of a simplified, efficient but not complete algorithm [31] The algorithm by Pugh was extended by Wonnacott and Pugh to handle arbitrary if conditions, arbitrary loop bounds and arbitrary array subscripts [33, 39, 40] The same ....

P. Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....To our knowledge, it is still an opened problem at this moment. But it gets more and more necessary to have new results in this research area, since the concept of parametric domains constitutes one of the major foundations in the theory of automatic parallelization of programs [16, 22]. In particular, it seems to be opportune to introduce the concept of multi dimensional homothetic systems and multi dimensional periodic numbers : this is the subject of our future works. Nevertheless, recall that our own method, previously presented in [7, 9, 8] also applied on the Gaussian ....

P. Feautrier. Parametric Integer Programming. RAIRO Recherche Op'erationnelle, 22:243268, September 1988.

....network is a jpj dimension grid . A combined cost function might even be better by taking advantage of the Manhattan distance to minimize the number of hops and of the lexicographical minimum to insure uniqueness. These problems can be cast as linear parametric problems and solved [35]. When no replication occurs, elementary data communications implied by Send Y and Receive Y can be parametrically enumerated in basis (p; u) where u is a basis for Y the local part of Y, the allocation and addressing of which are discussed in Section 4.3. Send and Receive are polyhedral sets ....

.... T :1; 0 a D:1; Li b 0 (n) Let Kx k 0 (n) be these constraints on x. Using (13) the constraints on u can be written K(x 0 (n; p) P u) k 0 (n) that is K 0 (n; p) where K = KP 0 Gammaa and k 0 (n; p) k 0 (n) Gamma Kx 0 (n; p) Algorithms presented in [4] or others [35, 30, 53, 2, 24, 23, 62, 59, 54, 86] can be used to generate the loop nest enumerating the local iterations. When S is of rank jaj, optimal code is generated because no projections are required. Otherwise, the quality of the control overhead depends on the accuracy of integer projections [73] but the correctness does not. ....

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Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

.... Successful parallelization may depend on the detection of subtle relations between program This research was supported in part by Centre de Recherche en Informatique de Montr eal, 1801 avenue McGill College, Bureau 800, Montr eal (Qu ebec) H3A 2N4 CANADA.E mail: jybertho crim.ca variables [Fea88] To be handled successfully, complicated subscripts may need program restructuring followed by a complex non linear analysis [Dum92] Large programs are built from many subroutines which have to be submitted to interprocedural dependence analysis [TIF86] a difficult process, or to inlining, ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....and not to create spurious ones. Finding the loop bounds of a reindexed loop nest is equivalent to scanning the integer points in an integer convex polyhedron. We use a parametrized version of the Dual Simplex method, implemented by Feautrier in a software called PIP (Parameter Integer Programming [9]) Much more details are given in [10] The PAF (Parall eliseur Automatique pour Fortran) automatic parallelizer designed by Feautrier follows the second way: dataflow analysis is performed [11] and an affine 3 schedule is derived for each statement, giving the execution date of every operation. ....

P. Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....network is a jpj dimension grid 4 . A combined cost function might even be better by taking advantage of the Manhattan distance to minimize the number of hops and of the lexicographical minimum to insure uniqueness. These problems can be cast as linear parametric problems and solved [12]. When no replication occurs, elementary data communications implied by send Y and receive Y can 2 This could be avoided by exchanging data first with processors p 0 such that p 0 p and then with processors such that p 0 p, using the lexicographic order. But this additional ....

Paul Feautrier. Parametric Integer Programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....that are usefull for compilation have to handle parametrized polyhedra defined by integer constraints. The PolyLib [10] handles parametrized rational polyhedra (image by affine function, convex hull, integer points count. Pips, Alpha and others rely on this library. On the other side, Pip [5] solves parametrized integer programming problems; it is used by several automatic parallelizers such as Paf [6] Bouclette [1] and Suif [7] Although the ODT are less expressive than the polyhedra (excepting the notion of modulo) they are enough to formalize the Array OL language. Moreover, our ....

Paul Feautrier. Parametric Integer Programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....is an opened problem since no other related results are known by us at this moment. But it gets more and more necessary to have new results in this research area, since the concept of parametric domains constitutes one of the major foundations in the theory of automatic parallelization of programs [17, 23]. In particular, it seems to be opportune to introduce the concept of multi dimensional homothetic systems and multi dimensional periodic numbers. Our geometrical model of a linear multiprocessor mapping is recalled in the next section. Associated with the above results, it allows the ....

P. Feautrier. Parametric Integer Programming. RAIRO Recherche Op'erationnelle, 22:243268, September 1988.

....lexicographical order. u k (l k ) stores the upper (lower) bound on the kth coordinate of scanned operations and is equal to the floor (ceiling) of the first component of the projection of D on the n Gamma k 1 first dimensions. u k and l k can be computed by thanks to software such as PIP [7]. If the upper bound is undefined, then u k is set equal to the corresponding placeholder (Section 2) When the current point corresponds to an actual execution, then last returns the corresponding cell in array A (passed as a second argument to last generator) If no scanned point corresponds to ....

P. Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....function of the original parameters. These new variables are ordered in the same order as letters of the alphabet. Compute the lexicographical maximum (u n ; un Gamma1 ; u 1 ) of (xn ; xn Gamma1 ; x 1 ) under the (affine) constraints A. This is exactly what a software such as PIP[10] was crafted for. The lexicographical maximum in M:P is C un n :C un Gamma1 n Gamma1 : C u1 1 Example: What is the lexicographical maximum of [a n i Gammaj ;b j ] 0in;0jn x 1 = n i Gamma j; x 2 = j ) lexmax(x 2 ; x 1 ) n; n) lexmax(M:P ) b n :a n . Lexicographical ....

P. Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....ZZ polyhedra is the question of their emptiness. It is a linear integer programming question which can be solved by the Gomory cut method which is integrated in the Parametric Integer Programming (PIP) tool. A straightforward application of PIP computes the lexicographic maximum of a ZZ polyhedron [5]. 2.1.2 Static Control Programs Static control programs are built from assignment statements and do loops. The only data structures are arrays of arbitrary dimensions. Loop bounds and array subscripts must be affine functions in the loop counters and integral structure parameters. An operation ....

P. Feautrier. Parametric integer programing. RAIRO Recherche op'erationnelle, 22:243268, Sept 1988.

....method is relevant here, but this method generates redundant constraints. An interesting alternative for this particular application is the Simplex method. A parametrized version of the Dual Simplex method has been developed by Feautrier and implemented in the PIP (Parameter Integer Programming [Fea88]) software. Unfortunately, the results produced by the PIP algorithm may sometimes be more intricate than necessary. In this paper, we show that the solution produced by PIP can be simplified. After setting the notations and recalling general definitions in Section 2, Section 3 illustrates some ....

....linear programming problems with two differences: ffl the cost function is the lexicographic order; ffl parameter values are generally unknown. PIP handles these two requirements 2 . We will briefly describe the PIP algorithm, but the interested reader will find a more complete description in [Fea88]. 2 PIP furthermore handles integer problems, thanks to the Gomory algorithm [Gom63] This part of the algorithm is not described here. 10 k j m = 1 n j=0 j = k m j = 1 k Figure 2: but not the maximum of these three. Suppose that we want the lexicographic minimum 3 of D( z) z is the ....

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P. Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....programs belongs to this class, and that many more may be converted to it by appropriate preprocessing. The analysis of addressing patterns in this class may be reduced to the solution of parametric systems of linear inequalities in integers, for which the author has devised an efficient algorithm [14]. The central problem to be solved here is the following: given an array cell, which of several statements is the source of the value contained therein at a given instant in the execution of a program. Most of the time, the statement will be embedded in a loop nest. Hence, we will require not ....

....evidence on the complexity of the algorithm and point to several possible extensions. The parametric integer algorithm, which is a basic component of the present method, is summarized in the appendix. For a more detailed presentation and proofs the reader is referred to the above quoted paper [14]. 2 1.2 Notations Bold letters will denote vectors or vector valued functions; jaj is the dimension of vector a. a[i: j] is the subvector of a built from components i to j. a[i] is a shorthand for a[i: i] Familiar operators and predicates like and will be tacitly extended to vectors. The ....

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Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988. 35

....may be reproduced (with suitable modifications) for all right hand side (rhs) references in any static control program. I have shown elsewhere [Fea91] that the process is completely mechanical, the basic tool being an algorithm for solving parametric linear programing problems in integers [Fea88]. The result of the analysis may be presented as the socalled Dataflow Graph or DFG for short. Table 1 gives the DFG of the Cholesky example 1 . The DFG has one node per statement in the original program. There is an edge from statement R (the source) to statement S (the sink) for each rhs ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....belongs to this class, and that many more may be converted to this form by appropriate preprocessing. The analysis of adressing patterns in this class may be reduced to the solution of parametric systems of linear inequalities in integers, for which we have devised an efficient algorithm in [Fea88b]. The first problem we will explore is the calculation of the data dependence relation. Beyond the construction of the dependence graph as used for instance by [Kuc78] or [AK82] our technique gives a precise criterium for loop interchange and allows the exact computation of direct dependences as ....

....existence predicate e r . ffl The elements of the feasible set are not ranked according to a linear cost function, as is customary, but according to the lexicographical order, The problem may be solved by an extension of Gomory s algorithm, the Parametric Integer Programming (PIP) algorithm of [Fea88b]. A short description may be found in the appendix; the algorithm may be adapted to cope with the computation of the lexical maximum, and for the elimination of one or more variables from a system of constraints. 3.3 Dependence vectors In the general case, the solution of a PIP is a multilevel ....

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Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....Q p (b) fu j f(u) g(b) 15) L s u h s 0; u[1: p] b[1: p] u[p 1] b[p 1]g; are linear equalities and inequalities, Q p (b) is the integer hull of a polyhedron. Finding its lexical maximum is a parametric integer problem, for which the author has devised an efficient algorithm in [Fea88b]. The parameters are the components of b and other integer variables (e.g. the variables which occurs in the array bounds) In the sequel, in the interest of legibility, we will not note explicitly these supplementary variables. Note also that the components of b are not arbitrary; they must ....

....(e.g. the variables which occurs in the array bounds) In the sequel, in the interest of legibility, we will not note explicitly these supplementary variables. Note also that the components of b are not arbitrary; they must satisfy various constraints, among which is: L t b h t 0: In [Fea88b] terminology, these inequalities form the context of the parametric integer problem. To express the solution, we need the concept of a quasi linear form. A quasilinear form is constructed from the parameters and integer constants by the operations of addition, multiplication by an integer, and ....

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Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....( if z = min D e0 ; e k (z) g 0 (z) else if z = min D e1 ; e k (z) f(z; v(max D j e1 ; e k (z Gamma e 0 ) Hence, we obtain the required new operator: Scan(D; e i ) i2f0; kg ; b; d; g 0 ) ffi OE . Practically, the problem (13) may be solved using the PIP software [5]. Indeed the definition of max D j e1 ; e k is based on a lexicographic maximum. Let us consider again the summation of the elements of a matrix. The corresponding system is: v2[i,j] case i,j i=1, j=1 : a[i,j] i,j 2 =i =n, j=1 : v2[i 1,n] a[i,j] i,j 1 =i =n, ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....expressions, and all structure parameters to integer variables. This restriction is enforced because at the present time our main analysis tool is the parametric linear integer 1 which also is the maximum value of the timing function. Preliminary Version, May 23, 1989 3 programming algorithm of [6]. It may be possible to extend the class of tractable programs by using formal computation techniques, see for instance [13] An important distinction when discussing parallel programs is the one between instructions and operations. An instruction is a static entity, which may be identified ....

....else (4, i, j) This reasoning may be reproduced (with suitable modifications) for all right hand side (rhs) references in the program. Whe have shown in [5] that the process is completely mechanical, the basic tool being an algorithm for solving parametric linear programing problems in integers ([6]) The corresponding program has been inplemented partly in Lisp and partly in C, and runs on various computers including a Dec Vax, a Sun workstation and a personnal computer. The result of the analysis may be presented as a graph which we call the data flow graph or DFG for short. The DFG has ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....may be reproduced (with suitable modifications) for all righthand side (rhs) references in any static control program. I have shown elsewhere [Fea91] that the process is completely mechanical, the basic tool being an algorithm for solving parametric linear programing problems in integers [Fea88] The corresponding program has been implemented partly in Lisp and partly in C, and runs on various computers including many workstations and a personal computer. The result of the analysis may be presented as the so called Dataflow Graph or DFG for short. Table 1 gives the DFG of the Cholesky ....

....vector of the source instruction in terms of the iteration vector of the sink instruction. In all practical cases we have encountered so far, this transformation is affine, but there is a possibility of encountering quasiaffine transformations as the result of the source computation (see [Fea88] for a somewhat artificial example) The source to sink transform of edge e will be denoted by h e . Its domain is P e , and its codomain is included in D oe(e) 1 Edges are numbered arbitrarily by the analysis software 6 The DFG is a synthetic representation of communication patterns in the ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....code has been structured in two layers: the outermost one implements the scheduling algorithm, the inner layer being a rudimentary algebraic calculator. The result is a parametric integer programming problem, which is sent to the PIP software. PIP is written in C and has been described elsewhere [7, 8]. 6.2 Some exemples The program in fig. 2 is a Fortran implementation of the familiar Gaussian elimination algorithm. The only unusual feature is the back substitution the ip loop in which subscripts have been rewritten in order to avoid negative increments. Table 2 is a list of the ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....to 1, which occurs very seldom, the resulting convex will be rational. But the approximation is acceptable since the rational convex will include P(z) There is no such an easy way to determine the vertex G(z) The more practicable method is to use a Parametric Integer Programming method (see [Fea88]) We use the software PIP to resolve the following problem: ff 0 = Minfff j ff 2 IN ; z ffK 2 Dg In this context G(z) z (ff 0 Gamma 1)K. Let us consider, now, a general multi dimensional reduction Reduc (D; l 1 ; l m ) b; d; g) For this reduction, the set P(z) can be ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....to 1, which occurs very seldom, the resulting convex will be rational. But the approximation is acceptable since the rational convex will include P(z) There is no such an easy way to determine the vertex G(z) The more practicable method is to use a Parametric Integer Programming method (see [Fea88]) We use the software PIP to resolve the following problem: ff 0 = Minfff j ff 2 IN; z ffK 2 Dg In this context G(z) z (ff0 Gamma 1)K. Let us consider, now, a general multi dimensional reduction Reduc (D; l 1 ; l m) b; d; g) For this reduction, the set P(z) can be expressed ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243-- 268, September 1988.

....to lexicographic order. We thus consider the following polyhedron F(k; m;n) F(k; m;n) f i; j j0 i m; 0 j n; 2i j = kg: 4) What is the lexicographical maximum of the integer valued vectors included in F(k; m;n) The aim of PIP is to solve such problems. The reader is refered to [1] for a mathematical description of the method. 1.1 General formulation Let F be a polyhedron: F( z) f xj x 0; A x B z c 0g: 5) In this formula, x is a vector with n entries: the vector of all unknowns. z, z 0, is the vector built from parameters and has p entries. Polyhedron F( z) ....

....bound on j can be found by solving the following problem: D 2 (k) f j; i ji m; j n; k i jg: This problem is to be solved in the context k m n. The input file may thus look like this: Lower bound on j after loop inversion (unknowns j i) parameters k m n) 2 3 3 1 1 1 ( #[0 1 0 0 1 0] #[ 1 0 0 0 0 1] #[1 1 0 1 0 0] #[ 1 1 1 0] The first sequence of integers should be read as: This problem has 2 unknowns (i and j) and 3 parameters (k, m and n) The domain is defined by 3 inequalities, the context by 1 inequality. There is no ( 1) big parameter and it is true (1) ....

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Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

....like Lisp. The code has been structured in two layers: the outermost one implements the scheduling algorithm, the inner layer being a rudimentary algebraic calculator. The result is a parametric linear program, which is sent to the PIP software. PIP is written in C and has been described elsewhere [7, 8]. 6.2 Some examples Consider again program 2. Table 2 is a list of the statements in the program, with a description of each iteration domain. Table 3 is a description of the associated Dataflow Graph. Edges are numbered arbitrarily by the software. An attempt to build a one dimensional schedule ....

Paul Feautrier. Parametric integer programming. RAIRO Recherche Op'erationnelle, 22:243--268, September 1988.

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