Maximum flow running time depend on longest path

Consider a directed graph with specified vertices s and t. The graph has m edges and n vertices with length of the longest simple st-path to be k. Dinic's algorithm finds a maximum flow consists of blocking flow computations, where each takes O(m\log n) time with dynamic trees [@Sleator1983; @Dinitz]. After each blocking flow, the distance between s and t in the residual graph increases by at least 1. Therefore there are at most k blocking flow computations. Dinic's algorithm takes O(km\log n) time to find a maximum st-flow.