Dijkstra's algorithm on directed acyclic graph with negative edges

Question

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]

Answer 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).