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By Utpal Banerjee

Dependence research might be thought of to be the second one version of the author's 1988 e-book, Dependence research for Supercomputing. it really is, even if, a totally new paintings that subsumes the fabric of the 1988 book. This booklet is the 3rd quantity in the sequence Loop changes for Restructuring Compilers. This sequence has been designed to supply a whole mathematical concept of changes that may be used to immediately swap a sequential software containing FORTRAN-like do loops into an identical parallel shape. In Dependence research, the writer extends the version to a application including do loops and project statements, the place the loops don't need to be sequentially nested and are allowed to have arbitrary strides. within the context of one of these software, the writer reports, intimately, dependence among statements of this system because of application variables which are components of arrays. Dependence research is directed towards graduate and undergraduate scholars, writers of restructuring compilers. The prerequisite for the booklet includes a few wisdom of programming languages, and familiarity with calculus and graph concept. No wisdom of linear programming is needed.

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Axo (s, t)U, integer Y0 The i n t e g e r m a t r i x U = ( b/gX°a/g ) is u n i m o d u l a r since d e t ( U ) = b)'o)/~l = 1. A n y 2-vector ( x , y ) can b e w r i t t e n as ( x , y ) = w h e r e (s, t) = ( x , y ) U -~. Since U is u n i m o d u l a r , ( x , y ) is an v e c t o r iff (s, t) is a n integer vector. Now, ( x , ) , ) is a s o l u t i o n CHAPTER 2. q~. If B d o e s n o t divide c, t h e n s is n o t an i n t e g e r a n d h e n c e (x, y ) c a n n o t b e an i n t e g e r vector.

C) If [ = ~ and S < T, then S ~ T. When checking for d e p e n d e n c e b e t w e e n two s t a t e m e n t s c a u s e d by a pair of p r o g r a m variables, we ignore all other variables and statem e n t s . Thus, we o f t e n abuse language a n d say "T d e p e n d s o n S" w h e n we k n o w m e r e l y that T d e p e n d s on S indirectly. 7), a n d t h e n partitioning the solution set into three subsets b a s e d o n w h e t h e r ~ < 3, or ~ > 3, or ~ = 3. 7 is applied to check for d e p e n d e n c e b e t w e e n s t a t e m e n t s in a single loop, c a u s e d by elem e n t s of a o n e - d i m e n s i o n a l array.

The s t a t e m e n t s S and T n e e d n o t be distinct, b u t C o n d i t i o n 2 requires that the instances S(i) a n d T(j) be distinct. Let ~ and 3 d e n o t e the iteration values c o r r e s p o n d i n g to i and j, respectively. , if i = j). In the l o o p - i n d e p e n d e n t case, we necessarily have S < T since S(~) is to be executed before T(i). Since a m e m o r y reference is either a "read" or a "write," a pair of s t a t e m e n t instances can reference the s a m e m e m o r y location in four different ways.

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