Network Flow and Testing Graph Connectivity

An algorithm of Dinic for finding the maximum flow in a network is described. It is then shown that if the vertex capacities are all equal to one, the algorithm requires at most $O(|V|^{1/2} \cdot |E|)$ time, and if the edge capacities are all equal to one, the algorithm requires at most $O(|V|^{2/3} \cdot |E|)$ time. Also, these bounds are tight for Dinic’s algorithm. These results are used to test the vertex connectivity of a graph in $O(|V|^{1/2} \cdot |E|^2 )$ time and the edge connectivity in $O(|V|^{5/3} \cdot |E|)$ time.