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<p>An approach to transformations for general loops in which dependence vectors represent precedence constraints on the iterations of a loop is presented. Therefore, dependences extracted from a loop nest must be lexicographically positive. This leads to a simple test for legality of compound transformations: any code transformation that leaves the dependences lexicographically positive is legal. The loop transformation theory is applied to the problem of maximizing the degree of coarse- or fine-grain parallelism in a loop nest. It is shown that the maximum degree of parallelism can be achieved by transforming the loops into a nest of coarsest fully permutable loop nests and wavefronting the fully permutable nests. The canonical form of coarsest fully permutable nests can be transformed mechanically to yield maximum degrees of coarse- and/or fine-grain parallelism. The efficient heuristics can find the maximum degrees of parallelism for loops whose nesting level is less than five.</p>
Index Termsparallel algorithm; loop iterations; coarse grain parallelism; wavefront; loop transformation theory; general loops; dependence vectors; precedence constraints; lexicographically positive; legality; compound transformations; code transformation; fine-grain parallelism; maximum degree; coarsest fully permutable loop nests; fully permutable nests; canonical form; heuristics; parallel algorithms; parallel programming; program compilers

M. Wolf and M. Lam, "A Loop Transformation Theory and an Algorithm to Maximize Parallelism," in IEEE Transactions on Parallel & Distributed Systems, vol. 2, no. , pp. 452-471, 1991.
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