Relations and Digraphs
Relations and digraphs (directed graphs) are ubiquitous in the the- ory of loop transformations. To study something as basic as the execution ordering of the iterations of a loop nest, we need to know about several ways of partially ordering integer vectors. The most important concept in loop transformations, namely that of depen- dence, can be viewed as a relation between the members of a certain set. This set could be the set of iterations of a loop nest, or the set of statements in the body of the nest, or the set of instances of the statements. We can even go to a lower level and talk about dependence between ‘loads’ and ‘stores,’ or to a higher level and discuss dependence between large parts of a program.
KeywordsEquivalence Relation Partial Order Maximal Element Direction Vector Minimal Element
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