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(n

^{2}).### 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);

printf(“\nEnter the adjacency matrix:\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);

}

}

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