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<p><b>Abstract</b>—This paper deals with the problem of transforming irregular data dependence structures of algorithms with nested loops into more regular ones. Algorithms under consideration are <it>n-dimensional</it> algorithms (algorithms with <it>n</it> nested loops) with <it>affine</it> dependences where dependences are affine functions of index variables of the loop. Methods are proposed to <it>uniformize</it> affine dependence algorithms, i.e., to transform affine dependence algorithms into <it>uniform dependence algorithms</it> where dependences are independent of the index variables (constant). Objectives are considered to guide the selection of feasible uniformizations. The first one is to reduce the number of dependences after uniformization. The second one is to maximize parallelism preserved by the uniformization. Some parallelism might be lost due to the uniformization. The parallelism preserved by the uniformization is measured by 1) the total execution time by the optimal <it>linear schedule</it> which assigns each computation in the algorithm an execution time according to a linear function of the index of the computation, and 2) the size of the cone spanned by the dependence vectors after uniformization.</p>
Affine dependence, communication localization, dependence transformation, nested loop, uniform dependence, uniformization.

W. Shang, Z. Chen and E. Hodzic, "On Uniformization of Affine Dependence Algorithms," in IEEE Transactions on Computers, vol. 45, no. , pp. 827-840, 1996.
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