Parallel transitive closure and transitive reduction algorithms

The authors provide distinct algorithms for computing transitive closure and transitive reduction with sequential time complexities of O(Σe_i) and O(n²+Σe_i), respectively, and parallel time complexities of O(e_i) And O(n +e_i), respectively. The proposed transitive closure algorithms are favorable for the sparse digraphs. It is still open whether transitive reduction may be computed in O(Σe _i) (or O(e_i) in parallel)