Dijkstra's algorithm on directed acyclic graph with negative edges

问题

Will Dijkstra's algorithm work on a graph with negative edges if it is acyclic (DAG)? I think it would because since there are no cycles there cannot be a negative loop. Is there any other reason why this algorithm would fail?

Thanks [midterm tomorrow]

回答1:

Consider the graph (directed 1 -> 2, 2-> 4, 4 -> 3, 1 -> 3, 3 -> 5):

  1---(2)---3--(2)--5
  |         |
 (3)      (2)
  |         |  
  2--(-10)--4

The minimum path is 1 - 2 - 4 - 3 - 5, with cost -3. However, Dijkstra will set d[3] = 2, d[2] = 3 in the first step, then extract node 3 from its priority queue and set d[5] = 4. Since node 3 was extracted from the priority queue, and Dijkstra does not push a given node to its priority queue more than once, it will never end up in it again, so the algorithm won't work.

Dijkstra's algorithm does not work with negative edges, period. The absence of a cycle changes nothing. Bellman-Ford is the one that can detect negative cost cycles and works with negative edges. Dijkstra will not work if you can have negative edges.

If you change Dijkstra's algorithm such that it can push a node to the priority queue more than once, then the algorithm will work with negative cost edges. But it is debatable if the new algorithm is still Dijkstra's: I would say you get Bellman-Ford that way, which is often implemented exactly like that (well, usually a FIFO queue is used and not a priority queue).

回答2:

I think Dijkstra's algorithm will work for DAG if there is no negative weight. Because Dijkstra's algorithm can't give the right answer for negative weighted edges graph. But sometimes it does based on graph type.

回答3:

Pure implementation of Dijkstra's will fail , whenever there is a negative edge weight. The following variant will still work for given problem scenario.

1. Every time an edge u -> v is relaxed, push a pair of (newer/shorter distance to v from source) into queue. This causes more than one copy of the same vertex in queue with different distances from source.
2. Continue to update the distance until queue is empty.

The above variant works, even if negative edges are present. But not in case if there is negative weight cycle. DAG is acyclic so, we don't have to worry about negative cycles.

There is more efficient way to calculate shortest path distances O(V+E) time for DAGs using topological ordering. More details can be found here

来源：https://stackoverflow.com/questions/28996844/dijkstras-algorithm-on-directed-acyclic-graph-with-negative-edges