Hybrid Bellman-Ford-Dijkstra algorithm

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