Hybrid Bellman-Ford-Dijkstra algorithm

We consider the single-source shortest paths problem in a digraph with negative edge costs allowed. A hybrid of the Bellman-Ford and Dijkstra algorithms is suggested, improving the running time bound of Bellman-Ford for graphs with a sparse distribution of negative cost edges. The algorithm iterates Dijkstra several times without re-initializing the tentative value d(v) at vertices. At most k+2 iterations solve the problem, if for any vertex reachable from the source, there exists a shortest path to it with at most k negative cost edges.In addition, a new, straightforward proof is suggested that the Bellman-Ford algorithm produces a shortest paths tree.