Question

Professor F. Lake suggests the following algorithm for finding the shortest path from node to node t in a directed graph with some negative edges: add a large constant to each edge weight so that all the weights become positive, then run Dijkstra’s algorithm starting at node s , and return the shortest path found to node t .

Is this a valid method? Either prove that it works correctly, or give a counterexample.

Short answer

Yes, this is a valid method.

Step-by-step solution

Explain Dijkstra’s algorithm

Dijkstra’s shortest-path algorithm marks all the distance of all the vertices as infinity. As the algorithm runs, when each vertex is visited, the distance between the vertices is calculated and the lowest distance values are updated at each iteration.

Is the given method is valid.

Consider the directed graph that has the nodes $s,p,q,t$ , The shortest path from node s to node t has to be calculated. Consider that some of the nodes from s to t have negative weight edges that is -2. Add the larger constant weight 5 to all the edges that makes the edge values as 3, that makes all the negative edges to positive edges.

Run Dijkstra’s algorithm and the shortest path will be returned.

Therefore, the given method is valid and it is proved