# Dijkstra Algorithm for Finding Shortest Path of a Graph

Dijkstra algorithm is also called single source shortest path algorithm. It is based on “greedy” technique. The algorithm maintains a list ‘visited[ ]’ of vertices, whose shortest distance from the source is already known.
If visited[1], equals 1, then the shortest distance of vertex i is already known. Initially, visited[i] is marked as, for source vertex.
At each step, we mark visited[v] as 1. Vertex v is a vertex at shortest distance from the source vertex. At each step of the algorithm, shortest distance of each vertex is stored in an array ‘distance[ ]’.

### Algorithm

1. Create cost matric C[ ][ ] from adjacency matrix adj[ ][ ]. C[i][j] is the cost of going from vertex i to vertex j. If there is no edge between vertices i and j then C[i][j] is infinity.
2. Array visited[ ] is initialized to zero.
for(i=0;i<n;i++)
visited[1]=0;
3. If the vertex 0 is the source vertex then visited[0] is marked as 1.
4. Create the distance matrix, by storing the cost of vertices from vertex no. 0 to n-1 from the source vertex 0.
for(i=1;i<n;i++)
distance[i]=cost[0][i];
Initial, distance of source vertex is taken as 0.
i.e. distance[0]=0;
5. for(i=1;i<n;i++)
– Choose a vertex w, such that distance[w] is minimum and visited[w] is 0.
Mark visited[w] as 1.
– Recalculate the shortest distance of remaining vertices from the source.
– Only, the vertices not marked as 1 in array visited[ ] should be considered for recalculation of distance.
i.e. for each vertex v
if(visited[v]==0)
distance[v]=min(distance[v],
distance[w]+cost[w][v])

### Time Complexity

The program contains two nested loops each of which has a complexity of O(n). n is number of vertices. So the complexity of algorithm is O(n2).

### C Program on Dijkstra Algorithm for Finding Minimum Distance of Vertices from a Given Source in a Graph

#include<stdio.h>
#include<conio.h>
#define INFINITY 9999
#define MAX 10
void dijkstra(int G[MAX][MAX],int n,int startnode);
void main()
{
int G[MAX][MAX],i,j,n,u;
printf(“Enter no. of vertices:”);
scanf(“%d”,&n);
for(i=0;i<n;i++)
for(j=0;j<n;j++)
scanf(“%d”,&G[i][j]);
printf(“\nEnter the starting node:”);
scanf(“%d”,&u);
dijkstra(G,n,u);
}
void dijkstra(int G[MAX][MAX],int n,int startnode)
{
int cost[MAX][MAX],distance[MAX],pred[MAX];
int visited[MAX],count,mindistance,nextnode,i,j;
//pred[] stores the predecessor of each node
//count gives the number of nodes seen so far
//create the cost matrix
for(i=0;i<n;i++)
for(j=0;j<n;j++)
if(G[i][j]==0)
cost[i][j]=INFINITY;
else
cost[i][j]=G[i][j];
//initialize pred[],distance[] and visited[]
for(i=0;i<n;i++)
{
distance[i]=cost[startnode][i];
pred[i]=startnode;
visited[i]=0;
}
distance[startnode]=0;
visited[startnode]=1;
count=1;
while(count<n-1)
{
mindistance=INFINITY;
//nextnode gives the node at minimum distance
for(i=0;i<n;i++)
if(distance[i]<mindistance&&!visited[i])
{
mindistance=distance[i];
nextnode=i;
}
//check if a better path exists through nextnode
visited[nextnode]=1;
for(i=0;i<n;i++)
if(!visited[i])
if(mindistance+cost[nextnode][i]<distance[i])
{
distance[i]=mindistance+cost[nextnode][i];
pred[i]=nextnode;
}
count++;
}
//print the path and distance of each node
for(i=0;i<n;i++)
if(i!=startnode)
{
printf(“\nDistance of node %d=%d”,i,distance[i]);
printf(“\n Path=%d”,i);
j=i;
do
{
j=pred[j];
printf(“<-%d”,j);
}while(j!=startnode);
}
}