Using selective path-doubling for parallel shortest-path computations

The author considers parallel shortest-path computations in weighted undirected graphs G=(V,E), where n= mod V mod and m= mod E mod . The standard path-doubling algorithms consists of O(log n) phases, where in each phase, for every triple of vertices (u/sub 1/, u/sub 2/, u/sub 3/) in V/sup 3/, she updates the distance between u/sub 1/ and u/sub 3/ to be no more than the sum of the previous-phase distances between (u/sub 1/, u/sub 2/) and (u/sub 2/, u/sub 3/). The work performed in each phase, O(n/sup 3/) (linear in the number of triples), is currently the bottleneck in NC shortest-paths computations. She introduces a new algorithm that for delta =o(n), considers only O(n delta /sup 2/) triples. Roughly, the resulting NC algorithm performs O(n delta /sup 2/) work and augments E with O(n delta ) new weighted edges such that between every pair of vertices, there exists a minimum weight path of size (number of edges) O(n/ delta ) (where O(f) identical to O(f polylog n)). To compute shortest-paths, she applies work-efficient algorithms, where the time depends on the size of shortest paths, to the augmented graph. She obtains a O(t) time O( mod S mod n/sup 2/+n/sup 3//t/sup 2/) work deterministic PRAM algorithm for computing shortest-paths form mod S mod sources to all other vertices, where t>