Difference between Prim's and Dijkstra's algorithms?

Question

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

Answer 1

@templatetypedef has covered difference between MST and shortest path. I've covered the algorithm difference in another So answer by demonstrating that both can be implemented using same generic algorithm that takes one more parameter as input: function f(u,v). The difference between Prim and Dijkstra's algorithm is simply which f(u,v) you use.

Answer 2

Directly from Dijkstra's Algorithm's wikipedia article:

The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual path.

Answer 3

Dijkstra's algorithm is a single source shortest path problem between node i and j, but Prim's algorithm a minimal spanning tree problem. These algorithm use programming concept named 'greedy algorithm'

If you check these notion, please visit

1. Greedy algorithm lecture note : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/07-greedy.pdf
2. Minimum spanning tree : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/20-mst.pdf
3. Single source shortest path : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/21-sssp.pdf

Answer 4

The simplest explanation is in Prims you don't specify the Starting Node, but in dijsktra you (Need to have a starting node) have to find shortest path from the given node to all other nodes.

Answer 5

At the code level, the other difference is the API.

You initialize Prim with a source vertex, s, i.e., Prim.new(s); s can be any vertex, and regardless of s, the end result, which are the edges of the minimum spanning tree (MST) are the same. To get the MST edges, we call the method edges().

You initialize Dijkstra with a source vertex, s, i.e., Dijkstra.new(s) that you want to get shortest path/distance to all other vertices. The end results, which are the shortest path/distance from s to all other vertices; are different depending on the s. To get the shortest paths/distances from s to any vertex, v, we call the methods distanceTo(v) and pathTo(v) respectively.

Answer 6

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!