Single-Source Shortest Paths, Nonnegative Weights

The single source shortest path algorithm (for non-negative weight) is also known Dijkstra algorithm. There is a given graph G(V,E) with its adjacency matrix representation, and a source vertex is also provided. Dijkstra’s algorithm to find the minimum shortest path between source vertex to any other vertex of the graph G.

From starting node to any other node, find the smallest distances. In this problem the graph is represented using the adjacency matrix. (Cost matrix and adjacency matrix is similar for this purpose).

Input − The adjacency matrix −

0 3 6 ∞ ∞ ∞ ∞
3 0 2 1 ∞ ∞ ∞
6 2 0 1 4 2 ∞
∞ 1 1 0 2 ∞ 4
∞ ∞ 4 2 0 2 1
∞ ∞ 2 ∞ 2 0 1
∞ ∞ ∞ 4 1 1 0

Output −

0 to 1, Using: 0, Cost: 3

0 to 2, Using: 1, Cost: 5

0 to 3, Using: 1, Cost: 4

0 to 4, Using: 3, Cost: 6

0 to 5, Using: 2, Cost: 7

0 to 6, Using: 4, Cost: 7

Algorithm

dijkstraShortestPath(n, dist, next, start)

Input − Total number of nodes n, distance list for each vertex, next list to store which node comes next, and the seed or start vertex.

Output − The shortest paths from start to all other vertices.

Begin
   create a status list to hold the current status of the selected node
   for all vertices u in V do
      status[u] := unconsidered
      dist[u] := distance from source using cost matrix
      next[u] := start
   done
   status[start] := considered, dist[start] := 0 and next[start] := φ
   while take unconsidered vertex u as distance is minimum do
      status[u] := considered
      for all vertex v in V do
         if status[v] = unconsidered then
         if dist[v] > dist[u] + cost[u,v] then
            dist[v] := dist[u] + cost[u,v]
            next[v] := u
      done
   done
End

Example(C++)

#include<iostream>
#define V 7
#define INF 999
using namespace std;
//Cost matrix of the graph
int costMat[V][V] = {
   {0, 3, 6, INF, INF, INF, INF},
   {3, 0, 2, 1, INF, INF, INF},
   {6, 2, 0, 1, 4, 2, INF},
   {INF, 1, 1, 0, 2, INF, 4},
   {INF, INF, 4, 2, 0, 2, 1},
   {INF, INF, 2, INF, 2, 0, 1},
   {INF, INF, INF, 4, 1, 1, 0}
};
int minimum(int *status, int *dis, int n){
   int i, min, index;
   min = INF;
   for(i = 0; i<n; i++)
      if(dis[i] < min && status[i] == 1){
      min = dis[i];
      index = i;
   }
   if(status[index] == 1)
      return index;//minimum unconsidered vertex distance
   else
      return -1;//when all vertices considered
}
void dijkstra(int n, int *dist,int *next, int s){
   int status[V];
   int u, v;
   //initialization
   for(u = 0; u<n; u++){
      status[u] = 1;//unconsidered vertex
      dist[u] = costMat[u][s];//distance from source
      next[u] = s;
   }
   //for source vertex
   status[s] = 2; dist[s] = 0; next[s] = -1;//-1 for starting vertex
   while((u = minimum(status, dist, n)) > -1){
      status[u] = 2;//now considered
      for(v = 0; v<n; v++)
         if(status[v] == 1)
            if(dist[v] > dist[u] + costMat[u][v]){
      dist[v] = dist[u] + costMat[u][v];//update distance
      next[v] = u;
      }
   }
}
main(){
   int dis[V], next[V], i, start = 0;
   dijkstra(V, dis, next, start);
   for(i = 0; i<V; i++)
      if(i != start)
         cout << start << " to " << i <<", Using: " << next[i] << ", Cost: " << dis[i] << endl;
}

Output

0 to 1, Using: 0, Cost: 3
0 to 2, Using: 1, Cost: 5
0 to 3, Using: 1, Cost: 4
0 to 4, Using: 3, Cost: 6
0 to 5, Using: 2, Cost: 7
0 to 6, Using: 4, Cost: 7