A matrix M indicating a minimum number of all optical paths between pairs of nodes may be generated in one embodiment using an algorithm for transitive closure. In various embodiments, different algorithms and methods may be used to generate matrix M. Once a convergent matrix M has been generated that attains transitive closure, any corresponding reachability matrix RM^a may be obtained from matrix M in a computationally efficient manner. Matrix M may be used to determine groups of potential regenerator placements and obtain end-to-end optical paths by selecting desired sequences of regenerators.