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Java Program to Find a Clique by Using the Technique of the Most Dense Subgraph

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  • Last Updated : 24 Dec, 2022
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A clique is a subset of vertices of a graph such that every two distinct vertices in the clique are adjacent. Finding a clique from a graph is an important problem in graph theory and many algorithms have been proposed to solve this problem. The most popular algorithm for finding a clique is the technique of the most dense subgraph. This technique is based on the fact that a clique is the most dense subgraph of the graph. In this article, we will discuss a Java program to find a clique by using the technique of the most dense subgraph.

Examples

Example 1:

Let’s take a graph G = (V, E) with the following adjacency matrix:

V = {v1, v2, v3, v4}

E = {(v1, v2), (v2, v3), (v3, v4), (v1, v4)}

Adjacency Matrix:

[[0, 1, 0, 1], 
[1, 0, 1, 0], 
[0, 1, 0, 1], 
[1, 0, 1, 0]]

The most dense subgraph of the graph G is the clique {v1, v2, v3, v4}.

Example 2:

Let’s take another graph G = (V, E) with the following adjacency matrix:

V = {v1, v2, v3, v4, v5}

E = {(v1, v2), (v2, v3), (v3, v4), (v4, v5), (v1, v5)}

Adjacency Matrix:

[[0, 1, 0, 0, 1], 
[1, 0, 1, 0, 0], 
[0, 1, 0, 1, 0], 
[0, 0, 1, 0, 1], 
[1, 0, 0, 1, 0]]

The most dense subgraph of the graph G is the clique {v1, v2, v3, v4}.

Approach

The approach used in the program is based on the technique of the most dense subgraph. The algorithm works as follows:

  • Initialize an array of size equal to the number of vertices in the graph.
  • For each vertex, count the number of edges connected to it.
  • Find the vertex with the maximum number of edges and mark it as the first vertex of the clique.
  • For each of the remaining vertices in the graph, count the number of edges connected to both the first vertex and the current vertex.
  • If the number of edges is equal to the number of vertices in the clique, then add the current vertex to the clique.
  • Repeat Steps 4 and 5 until all the vertices have been checked.

Below is the implementation of the above approach.

Implementation 

Java




// Java Program for the above approach
import java.util.ArrayList;
import java.util.List;
  
public class CliqueFinder {
  
    static int[][] adjMatrix;
  
    static int cliqueSize;
  
    static List<Integer> clique;
  
    // Function to initialize the adjacency matrix
    static void init(int[][] adjMatrix)
    {
        clique = new ArrayList<Integer>();
        for (int i = 0; i < adjMatrix.length; i++) {
            int count = 0;
            for (int j = 0; j < adjMatrix.length; j++) {
                if (adjMatrix[i][j] == 1)
                    count++;
            }
            if (count > cliqueSize) {
                cliqueSize = count;
                clique.clear();
                clique.add(i);
            }
        }
    }
  
    // Function to find the clique
    static void findClique(int[][] adjMatrix)
    {
        for (int i = 0; i < adjMatrix.length; i++) {
            if (!clique.contains(i)) {
                int count = 0;
                for (int j = 0; j < clique.size(); j++) {
                    if (adjMatrix[i][clique.get(j)] == 1)
                        count++;
                }
                if (count == cliqueSize) {
                    clique.add(i);
                }
            }
        }
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int[][] adjMatrix = { { 0, 1, 0, 1 },
                              { 1, 0, 1, 0 },
                              { 0, 1, 0, 1 },
                              { 1, 0, 1, 0 } };
  
        init(adjMatrix);
        findClique(adjMatrix);
  
        // Print the clique
        System.out.println("The clique is: ");
        for (int i = 0; i < clique.size(); i++) {
            System.out.print(clique.get(i) + " ");
        }
    }
}


Output

The clique is: 
0 

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