Automatic coarse-grained parallelization of program loops is of great importance for multi-core computing systems. This paper presents a comparison of Iteration Space Slicing and Affine Transformation Framework algorithms aimed at extracting coarse-grained parallelism available in arbitrarily nested parameterized affine loops. We demonstrate that Iteration Space Slicing permits for extracting more coarse-grained parallelism in comparison to the Affine Transformation Framework. Experimental results show that by means of Iteration Space Slicing algorithms, we are able to extract coarse-grained parallelism for most loops of the NAS and UTDSP benchmarks, and that there is a strong need in devising advanced algorithms for calculating the exact transitive closure of dependence relations in order to increase the applicability of that framework.
The set of paths in a graph is an important concept with many applications in system analysis. In the context of integer tuple relations, which can be used to represent possibly infinite graphs, this set corresponds to the transitive closure of the relation representing the graph. Relations described using only affine constraints and projection are fairly efficient to use in practice and capture Presburger arithmetic. Unfortunately, the transitive closure of such a quasi-affine relation may not be quasi-affine and so there is a need for approximations. In particular, most applications in system analysis require overapproximations. Previous work has mostly focused either on underapproximations or special cases of affine relations. We present a novel algorithm for computing overapproximations of transitive closures for the general case of quasi-affine relations (convex or not). Experiments on non-trivial relations from real-world applications show our algorithm to be on average more accurate and faster than the best known alternatives.
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