NC ALGORITHMS FOR DYNAMICALLY SOLVING THE ALL PAIRS SHORTEST PATHS PROBLEM AND RELATED PROBLEMS

We consider the problem of dynamically maintaining a solution of all p airs shortest paths in a directed weighted graph G = (V, E) undergoing a sequence of edge insertions and/or the edge cost decreases, and pre sent a simple data structure to support the above operations, The prop osed algorithm for maintaining the data structure requires O(log n) ti me and O(n(2)) processors for each of the operations above. Furthermor e, our algorithm is able to answer n(2) queries concerning the lengths of all pairs shortest paths in O(1) time, and to find n shortest path s in O(log n) time. The same bounds for similar operations can be achi eved for other problems such as dynamically maintaining all pairs long est paths in a directed acyclic graph (DAG),the topological order of a DAG, and the transitive closure of a directed graph. To the best of o ur knowledge, no partially dynamic NC algorithm using O(n(2)) processo rs for these problems is known yet, despite the existence of several e fficient sequential algorithms, All known NC algorithms for these prob lems are based on matrix multiplication which requires M(n) processors at least. Currently the best result for M(n) is n(2.376) (Coppersmith and Winograd, 1987). Unless otherwise specified, our computational mo del is a CREW PRAM in which concurrent read to the same memory cell is allowed, but concurrent write to the same memory cell is forbidden.