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[ ]’.

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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);

    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);

            }

}