Question

I have been fighting all day in understanding Dijkstra’s algorithm and implementing with no significant results. I have a matrix of cities and their distances. What I want to do is to given an origin point and a destination point, to find the shortest path between the cities.

Example:

     __0__ __1__ __2__ 
 0  |  0  | 34  |  0  |
    |-----|-----|-----|
 1  | 34  |  0  | 23  |
    |-----|-----|-----|
 2  |  0  | 23  |  0  |
     ----- ----- -----

I started wondering if there is an other way to solve this. What if I apply Prim’s algorithm from the origin’s point and then I loop through the whole tree created until I find the destination point?

Answer 1

You could apply Prim’s algorithm and then walk the resulting tree, but you answer may be wrong. Assume that you have a graph where each edge has the same weight. Prim’s algorithm simply chooses a minimal weight edge in the set of edges that could be added to the tree. It is possible that you will not choose an edge that will lead to a shortest path between two nodes.
Assume:

     __0__ __1__ __2__ 
 0  |  0  |  1  |  1  |
    |-----|-----|-----|
 1  |  1  |  0  |  1  |
    |-----|-----|-----|
 2  |  1  |  1  |  0  |
     ----- ----- -----

Starting from node 0 you could, via Prim’s, choose the edges 0-1 and 0-2 to make your tree. Alternately, you could pick edges 0-1 and 1-2 to make your tree. Under the first edge set, you could find the minimum length path from 0 to 2, but under the second edge set you would not find the minimal path. Since you can’t a-priori determine which edges get added in the Prim algorithm, you can’t use it to find a shortest path.

You could consider the Bellman-Ford algorithm, but unless you’re dealing with negative edge weights I find Dijkstra’s algorithm preferable.