Difference between Prim's and Dijkstra's algorithms?

What is the exact difference between Dijkstra's and Prim's algorithms? I know Prim's will give a MST but the tree generated by Dijkstra will also be a MST. Then what is the exact difference?

Solution 1:

Prim's algorithm constructs a minimum spanning tree for the graph, which is a tree that connects all nodes in the graph and has the least total cost among all trees that connect all the nodes. However, the length of a path between any two nodes in the MST might not be the shortest path between those two nodes in the original graph. MSTs are useful, for example, if you wanted to physically wire up the nodes in the graph to provide electricity to them at the least total cost. It doesn't matter that the path length between two nodes might not be optimal, since all you care about is the fact that they're connected.

Dijkstra's algorithm constructs a shortest path tree starting from some source node. A shortest path tree is a tree that connects all nodes in the graph back to the source node and has the property that the length of any path from the source node to any other node in the graph is minimized. This is useful, for example, if you wanted to build a road network that made it as efficient as possible for everyone to get to some major important landmark. However, the shortest path tree is not guaranteed to be a minimum spanning tree, and the sum of the costs on the edges of a shortest-path tree can be much larger than the cost of an MST.

Another important difference concerns what types of graphs the algorithms work on. Prim's algorithm works on undirected graphs only, since the concept of an MST assumes that graphs are inherently undirected. (There is something called a "minimum spanning arborescence" for directed graphs, but algorithms to find them are much more complicated). Dijkstra's algorithm will work fine on directed graphs, since shortest path trees can indeed be directed. Additionally, Dijkstra's algorithm does not necessarily yield the correct solution in graphs containing negative edge weights, while Prim's algorithm can handle this.

Hope this helps!

Solution 2:

Dijkstra's algorithm doesn't create a MST, it finds the shortest path.

Consider this graph

       5     5
  s *-----*-----* t
     \         /
       -------
         9

The shortest path is 9, while the MST is a different 'path' at 10.

Solution 3:

Prim and Dijkstra algorithms are almost the same, except for the "relax function".

Prim:

MST-PRIM (G, w, r) {
    for each key ∈ G.V
        u.key = ∞
        u.parent = NIL
    r.key = 0
    Q = G.V

    while (Q ≠ ø)
        u = Extract-Min(Q)
        for each v ∈ G.Adj[u]
            if (v ∈ Q)
                alt = w(u,v)    <== relax function, Pay attention here
                if alt < v.key
                    v.parent = u
                    v.key = alt
}

Dijkstra:

Dijkstra (G, w, r) {
    for each key ∈ G.V
        u.key = ∞
        u.parent = NIL
    r.key = 0
    Q = G.V

    while (Q ≠ ø)
        u = Extract-Min(Q)
        for each v ∈ G.Adj[u]
            if (v ∈ Q)
                alt = w(u,v) + u.key  <== relax function, Pay attention here
                if alt < v.key
                    v.parent = u
                    v.key = alt
}

The only difference is pointed out by the arrow, which is the relax function.

• The Prim, which searches for the minimum spanning tree, only cares about the minimum of the total edges cover all the vertices. The relax function is alt = w(u,v)
• The Dijkstra, which searches for the minimum path length, so it cares about the edge accumulation. The relax function is alt = w(u,v) + u.key

Solution 4:

Dijsktra's algorithm finds the minimum distance from node i to all nodes (you specify i). So in return you get the minimum distance tree from node i.

Prims algorithm gets you the minimum spaning tree for a given graph. A tree that connects all nodes while the sum of all costs is the minimum possible.

So with Dijkstra you can go from the selected node to any other with the minimum cost, you don't get this with Prim's