<p>You are given an <code>n x n</code> binary matrix <code>grid</code>. You are allowed to change <strong>at most one</strong> <code>0</code> to be <code>1</code>.</p>

<p>Return <em>the size of the largest <strong>island</strong> in</em> <code>grid</code> <em>after applying this operation</em>.</p>

<p>An <strong>island</strong> is a 4-directionally connected group of <code>1</code>s.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1,0],[0,1]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> Change one 0 to 1 and connect two 1s, then we get an island with area = 3.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1,1],[1,0]]
<strong>Output:</strong> 4
<strong>Explanation: </strong>Change the 0 to 1 and make the island bigger, only one island with area = 4.</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1,1],[1,1]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> Can&#39;t change any 0 to 1, only one island with area = 4.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= n &lt;= 500</code></li>
	<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
</ul>

class Solution {
public:
    // Function to find the parent with path compression
    int findp(int s, unordered_map<int, int>& parent) {
        if (parent[s] != s) {
            parent[s] = findp(parent[s], parent);  // Path compression
        }
        return parent[s];
    }

    // Function to perform union by rank
    void unionf(int a, int b, unordered_map<int, int>& parent, unordered_map<int, int>& rank) {
        int pa = findp(a, parent);
        int pb = findp(b, parent);

        if (pa == pb) return;

        // Union by rank
        if (rank[pa] < rank[pb]) {
            parent[pa] = pb;
        } else if (rank[pa] > rank[pb]) {
            parent[pb] = pa;
        } else {
            parent[pa] = pb;
            rank[pb]++;
        }
    }

    int largestIsland(vector<vector<int>>& grid) {
        unordered_map<int, int> parent;
        unordered_map<int, int> rank;
        int n = grid.size();
        int maxSize = 0;

        // Initialize parent and rank
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                parent[i * n + j] = i * n + j;
                rank[i * n + j] = 1;
            }
        }

        // Perform union-find for all 1s
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 1) {
                    int dr[] = {-1, 0, 1, 0};  // Directions for rows
                    int dc[] = {0, -1, 0, 1};  // Directions for columns
                    for (int l = 0; l < 4; l++) {
                        int nrow = i + dr[l];
                        int ncol = j + dc[l];
                        if (nrow >= 0 && nrow < n && ncol >= 0 && ncol < n && grid[nrow][ncol] == 1) {
                            unionf(i * n + j, nrow * n + ncol, parent, rank);
                        }
                    }
                }
            }
        }

       
        unordered_map<int, int> area;  
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 1) {
                    int root = findp(i * n + j, parent);
                    area[root]++;
                    maxSize = max(maxSize, area[root]);  
                }
            }
        }

      
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 0) {
                    int dr[] = {-1, 0, 1, 0};
                    int dc[] = {0, -1, 0, 1};
                    unordered_set<int> uniqueIslands;
                    int potentialSize = 1; 

                    for (int l = 0; l < 4; l++) {
                        int nrow = i + dr[l];
                        int ncol = j + dc[l];
                        if (nrow >= 0 && nrow < n && ncol >= 0 && ncol < n && grid[nrow][ncol] == 1) {
                            int root = findp(nrow * n + ncol, parent);
                            if (uniqueIslands.find(root) == uniqueIslands.end()) {
                                potentialSize += area[root];
                                uniqueIslands.insert(root);
                            }
                        }
                    }

                    maxSize = max(maxSize, potentialSize);
                }
            }
        }

        return maxSize;
    }
};

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