By Utpal Banerjee
Automatic transformation of a sequential software right into a parallel shape is a topic that provides a good highbrow problem and supplies an outstanding functional award. there's a great funding in latest sequential courses, and scientists and engineers proceed to jot down their software courses in sequential languages (primarily in Fortran). The call for for larger speedups raises. The task of a restructuring compiler is to find the dependence constitution and the features of the given computer. a lot recognition has been excited by the Fortran do loop. this is often the place one expects to discover significant chunks of computation that must be played many times for various values of the index variable. Many loop changes were designed through the years, and a number of other of them are available in any parallelizing compiler presently in use in or at a school study facility.
The booklet sequence on KappaLoop variations for Restructuring Compilerskappa offers a rigorous conception of loop adjustments and dependence research. we wish to advance the differences in a constant mathematical framework utilizing gadgets like directed graphs, matrices, and linear equations. Then, the algorithms that enforce the differences will be accurately defined when it comes to sure summary mathematical algorithms. the 1st quantity, LoopTransformations for Restructuring Compilers: The Foundations, supplied the final mathematical historical past wanted for loop adjustments (including these simple mathematical algorithms), mentioned info dependence, and brought the most important modifications. the present quantity, Loop Parallelization, builds an in depth conception of iteration-level loop variations in accordance with the fabric built within the past ebook.
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A dependence at level u before the interchange becomes a dependence at level v after the interchange, where u if 1 :S u < p v = { p or p + 1 if u = p p ifu = p + 1 u if p + 1 < u :S m. Dependence at level u is due to one or more direction vectors of the form (0,0, ... ,0,1,*,*, ... ,*) with (u -1) leading zeros. If 1 ::; u < p or p + 1 < u :S m, then the interchange of loops Lp and Lp+ 1 does not change the first u elements of this direction vector form. Hence, a dependence at such a level u will still be a dependence at the same level after the interchange.
Consider next the question of equivalence of L1' to L. 1. 1. (i 1 ,i 2 ) -< (jl,j2), since H(i 1,i 2) must execute before H(jl,j2) in L. The condition for the equivalence of Lp to L requires that H(i 1, i 2 ) execute before H(jl, j2) also in L p , that is, (i 2, id -< (j2, jd. The only situation where (iI, i 2) -< (jl,j2) holds and (i 2, i 1) -< (j2,jl) does not is when il < jl and i2 > j2. Thus, interchange ofthe loops in L is invalid iff there is an iteration H(jl,j2) depending on an iteration H(i 1 , i 2) such that il < jl and i2 > j2.
This implies dP = jP - iP >- O. 0 Corollary 1 If the permutation of L defined by the matrix P zs valid, then the distance matrix of the transformed program Lp zs VP where V is the distance matrix of L. PROOF. The proof is left as an exercise. o 36 CHAPTER 2. LOOP PERMUTATIONS One very important feature of loop permutations is that their validity can be determined from a knowledge of direction vectors alone. This is quite useful since the number of direction vectors is never more (and often less) than the number of distance vectors, and direction vectors may be easier to compute and store than distance vectors.