Coarse-Grained Parallel Transitive Closure Algorithm: Path Decomposition Technique

We investigate the relation between fine-grained and coarse-grained distributed computations of a class of problems related to the generic transitive closure problem (TC for short). We choose an intricate systolic algorithm for the TC problem, by Guibas, Kung and Thompson (GKT algorithm for short), as a starting point due to its particularly close relationship to matrix multiplication. The GKT algorithm reduces the TC problem to three successive parallel matrix multiplications. We extract the main ideas of this algorithm, namely different path decompositions related to min-paths and max-paths computations and devise a two-pass parallel algorithm, such that the second pass is purely a triangular matrix multiplication involving exactly 3 of the total number of elementary operations (multiplying two single elements of the matrix). This is helpful in coarsegrained parallel computations since matrix multiplication is well parallelizable. A novel approach is used and as a first result a more efficient and simpler two-pass fine-grained algorithm is designed. The second result is a non-trivial transformation of this fine-grained algorithm into a coarse-grained (and more practical) version. The full proof of correctness of the transformation, which is presented in the appendices, is quite complex and is the hardest result of the paper. Our algorithms are specially structured to directly show the correspondence between the main fine-grained and the main coarse-grained operations.