Dijkstra confusion

 Dijkstra is a famous algorithm for the shortest path problem. The original paper is from 1959 [1]:

1. Dijkstra just does not work correctly is there are negative weights
2. Dijkstra does not work if there are negative cycles in the graph
3. Keep things vague and mention negative weights and negative cycles

Reason 2 would imply that a problem with negative weights would work as long as there are no negative cycles.

Looking at questions about this on stackexchange, I see there is much confusion about this. Also, I see vague, misleading messages in documentation and error/warning messages. For instance, lets have a look at [2]:

Also, this routine does not work for graphs with negative distances. Negative distances can lead to infinite cycles that must be handled by specialized algorithms such as Bellman-Ford’s algorithm or Johnson’s algorithm.

Well, this is sufficiently vague. We deal with option 3: we don't know if this is referring to reason 1 or 2. The same function can produce a warning:

UserWarning: Graph has negative weights: dijkstra will give inaccurate results if the graph contains negative cycles. Consider johnson or bellman_ford.
  distmat,pred = scipy.sparse.csgraph.dijkstra(spmat,indices=source,return_predecessors=True)

This seems to say we only need to worry about graphs with negative cycles. This seems to be option 2.

A small counter example

In [3] a small acyclic graph (a DAG: Directed Acyclic Graph) is presented:

dijkstra
shortest path length: 1.0
johnson
shortest path length: -8.0
bellman_ford
shortest path length: -8.0

<ipython-input-25-7a1de3894d98>:13: UserWarning: Graph has negative weights: dijkstra will give inaccurate results if the graph contains negative cycles. Consider johnson or bellman_ford.
  distmat,pred = scipy.sparse.csgraph.dijkstra(spmat,indices=source,return_predecessors=True)

This example disproves that we just have to watch out for negative cycles. 

Conclusion

Dijkstra [2] does not work with negative weights or lengths. Forget about negative cycles: that is not a useful check. It would be better if documentation and warning messages are not implying that we need to look at negative cycles. 

To make things more complicated: there are implementations of Dijkstra's algorithm that actually work with negative weights [4].

References

1. Dijkstra, E.W. A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959).
2. https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csgraph.dijkstra.html
3. https://stackoverflow.com/questions/53532109/can-i-use-dijkstras-algorithm-on-dag-with-negative-weighted-edges
4. Robert Sedgewick and Kevin Wayne, Algorithms, 4th Edition, https://algs4.cs.princeton.edu/44sp/ See the question: Does Dijkstra's algorithm work with negative weights?