Two-Levels-Greedy: a generalization of Dijkstra's shortest path algorithm

The shortest path problem on weighted directed graphs is one of the basic network optimization problems. Its importance is mainly due to its applications in various areas, such as communication and transportation. Here we are interested in the single-source case. When the graph is not required to satisfy any particular restriction and negative weight edges can occur, the problem is solved by the Bellman-Ford-Moore algorithm [Bel58,For56,Moo59], whose complexity is O(|V ||E|), with V and E denoting the sets of nodes and of edges, respectively. A more efficient solution due to Dijkstra [Dij59] is available when weights are restricted to non-negative values. Depending on the implementation used for maintaining a service priority queue, Dijkstra’s algorithm has complexity O(|V |2) (simple list), or O(|E| log |V |) (standard binary heap), or O(|V | log |V | + |E|) (Fibonacci heap [FT87]). Another case which can be solved very efficiently occurs when the underlying graph is acyclic. In such a case, by scanning the nodes in topological ordering, one can achieve a O(|V |+ |E|) complexity. In this note we present a natural generalization of Dijkstra’s algorithm to the case in which negative weight edges are allowed, but only outside of any cycle. The resulting algorithm turns out to have the same asymptotic complexity of Dijkstra’s algorithm and shows a linear behavior in the case of acyclic graphs. In fact, we will also see that our proposed algorithm compares