Representing a graph with a 2D array

Question

I have a question where I represent a graph in terms of a 2D array. I have a sample as well, but I have no idea, how it works.... This is the graph I am given

And this is how they represent it using a 2D array

How does one translate to the other?

Also, this is a part of an implementation of Dijsktra's algorithm. Here is the code for your reference, it is taken from geeksforgeeks

// A Java program for Dijkstra's single source shortest path algorithm.
// The program is for adjacency matrix representation of the graph
import java.util.*;
import java.lang.*;
import java.io.*;

class ShortestPath {
    // A utility function to find the vertex with minimum distance value,
    // from the set of vertices not yet included in shortest path tree
    static final int V = 9;
    int minDistance(int dist[], Boolean sptSet[])
    {
        // Initialize min value
        int min = Integer.MAX_VALUE, min_index = -1;

        for (int v = 0; v < V; v++)
            if (sptSet[v] == false && dist[v] <= min) {
                min = dist[v];
                min_index = v;
            }

        return min_index;
    }

    // A utility function to print the constructed distance array
    void printSolution(int dist[])
    {
        System.out.println("Vertex \t\t Distance from Source");
        for (int i = 0; i < V; i++)
            System.out.println(i + " \t\t " + dist[i]);
    }

    // Function that implements Dijkstra's single source shortest path
    // algorithm for a graph represented using adjacency matrix
    // representation
    void dijkstra(int graph[][], int src)
    {
        int dist[] = new int[V]; // The output array. dist[i] will hold
        // the shortest distance from src to i

        // sptSet[i] will true if vertex i is included in shortest
        // path tree or shortest distance from src to i is finalized
        Boolean sptSet[] = new Boolean[V];

        // Initialize all distances as INFINITE and stpSet[] as false
        for (int i = 0; i < V; i++) {
            dist[i] = Integer.MAX_VALUE;
            sptSet[i] = false;
        }

        // Distance of source vertex from itself is always 0
        dist[src] = 0;

        // Find shortest path for all vertices
        for (int count = 0; count < V - 1; count++) {
            // Pick the minimum distance vertex from the set of vertices
            // not yet processed. u is always equal to src in first
            // iteration.
            int u = minDistance(dist, sptSet);

            // Mark the picked vertex as processed
            sptSet[u] = true;

            // Update dist value of the adjacent vertices of the
            // picked vertex.
            for (int v = 0; v < V; v++)

                // Update dist[v] only if is not in sptSet, there is an
                // edge from u to v, and total weight of path from src to
                // v through u is smaller than current value of dist[v]
                if (!sptSet[v] && graph[u][v] != 0 && dist[u] != Integer.MAX_VALUE && dist[u] + graph[u][v] < dist[v])
                    dist[v] = dist[u] + graph[u][v];
        }

        // print the constructed distance array
        printSolution(dist);
    }

    // Driver method
    public static void main(String[] args)
    {
        /* Let us create the example graph discussed above */
        int graph[][] = new int[][] { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },
                                    { 4, 0, 8, 0, 0, 0, 0, 11, 0 },
                                    { 0, 8, 0, 7, 0, 4, 0, 0, 2 },
                                    { 0, 0, 7, 0, 9, 14, 0, 0, 0 },
                                    { 0, 0, 0, 9, 0, 10, 0, 0, 0 },
                                    { 0, 0, 4, 14, 10, 0, 2, 0, 0 },
                                    { 0, 0, 0, 0, 0, 2, 0, 1, 6 },
                                    { 8, 11, 0, 0, 0, 0, 1, 0, 7 },
                                    { 0, 0, 2, 0, 0, 0, 6, 7, 0 } };
        ShortestPath t = new ShortestPath();
        t.dijkstra(graph, 0);
    }
}

If this is the graph given to me, for example, how would I represent it using a 2D array?

Answer 1

You can read it like a coordinate table. Every row represents a single vertex and every column value on the same row represents the distance to the Nth vertex.

Examples:

0, 1: first row, second column has the value of 4; which means vertex 0's distance to vertex 1 is 4.

1, 7: second row, 8th column has the value of 11; which means vertex 1's distance to vertex 7 is 11.

And so on...

Answer 2

They have used an adjacency matrix to represent an weighted undirected graphs. According to geeksforgeeks:

Adjacency Matrix: Adjacency Matrix is a 2D array of size V x V where V is the 
number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 
indicates that there is an edge from vertex i to vertex j. Adjacency matrix for 
undirected graph is always symmetric. Adjacency Matrix is also used to represent 
weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex 
j with weight w.  

As the last two line state, adjacency matrix can be used to store a weighted graph which they have done in this case.

There might be few problems if you use adjacency matrix to represent weighted graphs. Here we use 0 to represent no-edge between any two vertices. If there is any graph which has a weight 0in the graph, then a little change in the adjacency matrix will be needed since 0 already represents no-edge.

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In the comments you mention about another graph. That one, however, is not weighted. It is also not undirected graph since its not symmetric.

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Do comment if you have any other doubts.

Answer 3

I think I've figured it out!

In the example graph, there are 9 vertices. Therefore a 9x9 2D matrix is created, such that:

• Row 1 corresponds to vertex 0, Row 2 corresponds to vertex 1 and so on

and

• Column 1 corresponds to vertex 0, Column 2 corresponds to vertex 1 and so on

and the array is filled such that it contains the weight of the edges between node x and y. For example: Row 1 column 2 (map0) contains the weight of the edge between vertex 0 and vertex 1.

Therefore for a graph with n vertices, an nxn array needs to be created such that the location[x][y] in the array contains the weight between edge from x to y.

As for the example graph I had asked the solution for, this would be its corresponding 2D Matrix: It is a little big, because we create a 16x16 2D matrix