## Abstract

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.

Original language | English |
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Pages (from-to) | 35-44 |

Number of pages | 10 |

Journal | Journal of Discrete Algorithms |

Volume | 42 |

DOIs | |

State | Published - 1 Jan 2017 |

## Keywords

- Algorithm
- Graph
- Negative edge
- Shortest path