I am trying to solve the following problem:

Input is $2D$ array of integers, $M$, which corresponds to friendship relations. For example, if $M[1][2]=1$, $1$ and $2$ are friends (assuming symmetry it is also true that $M[2][1]=1$). If $M[2][3]=1$, then $\{1,2,3\}$ is a community. If all other entries are $0$, then $\{0\}$ is a community by itself, $\{4\}$ is a community by itself, etc. One can think of $M$ as representation of a graph with $V$ nodes, and $E$ edges. $M$ is $VxV$ matrix and the entries with value $1$ correspond to edges of an undirected graph. For the above example, the total number of communities is $V-2$.

I need to explain the following:

If we solve the above problem with union and find operations (weighted quick union), it will take $O(E+Vlog(V))$ time, explain.

I wrote the code for this. I am also including weighted union and find algorithm code as well below for completeness (but it is a standard known algorithm I guess). With the way I implement it, it does not look like this algorithm takes O(E+Vlog(V)) time. I need to understand how can this be done in O(E+Vlog(V)) time.

It is said that find operation takes O(log(V)) time because the tree is almost balanced with weighted quick union algorithm.

This is my code: enter image description here

The code for weighted quick union and find (given):

public class WeightedQuickUnion {

int[] parent;   // parent[i] = parent of i  
int[] size;     // size[i] = number of sites in subtree rooted at i
int count;      // number of components

 * Initializes an empty union–find data structure 
 * Each site is initially in its own 
 * component.
 * n is the number of sites
public WeightedQuickUnion(int n) {
    count = n;
    parent = new int[n];
    size = new int[n];
    for (int i = 0; i < n; i++) {
        parent[i] = i;
        size[i] = 1;

 * Return the number of components.
public int count() {
    return count;

 * return the component identifier for the component containing p.
 * p the integer representing the item
public int find(int p) {
    while (p != parent[p])
        p = parent[p];
    return p;

// validate that p is a valid index
private void validate(int p) {
    int n = parent.length;
    if (p < 0 || p >= n) {
        throw new IllegalArgumentException("index " + p + " is not between 0 and " + (n-1));  

 * return true if the the item p and item q are in the same component.
public boolean connected(int p, int q) {
    return find(p) == find(q);

 * Merge the component containing p with the 
 * the component containing q.
public void union(int p, int q) {
    int rootP = find(p);
    int rootQ = find(q);

    if (rootP == rootQ) return;

    // make smaller root point to larger one
    if (size[rootP] < size[rootQ]) {
        parent[rootP] = rootQ;
        size[rootQ] += size[rootP];
    else {
        parent[rootQ] = rootP;
        size[rootP] += size[rootQ];

You are using Adjacency matrix representation, instead just maintain a list of pairs of friend (i.e. edge list) and iterate it once.


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