<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 DisjointSet {
public:
    vector<int> parent, size;
    DisjointSet(int v) {
        parent.resize(v + 1);
        size.resize(v + 1);
        for (int i = 0; i <= v; i++) {
            parent[i] = i;
            size[i] = 1;
        }
    }

    int findParent(int node) {
        if (node == parent[node]) {
            return node;
        }
        return parent[node] = findParent(parent[node]);
    }

    void Union(int u, int v) {
        int par_u = findParent(u);
        int par_v = findParent(v);

        if (par_u == par_v)
            return;

        if (size[par_u] < size[par_v]) {
            parent[par_u] = par_v;
            size[par_v] += size[par_u];
        } else {
            parent[par_v] = par_u;
            size[par_u] += size[par_v];
        }
    }
};

class Solution {
public:
    int largestIsland(vector<vector<int>>& grid) {
        int n = grid.size();
        int m = grid[0].size();
        DisjointSet ds(n * m + 1);
        int ans = 0;
        int dir[] ={0, 1, 0, -1};
        
        for(int i = 0; i < n; i++){
            for(int j = 0; j < m; j++){
                if(grid[i][j]){
                    int curr = i * m + j;
                    for(int t = 0; t < 4; t++){
                        int x = i + dir[t], y = j + dir[3-t];
                        if(x >= 0 && y >= 0 && x < n && y < m && grid[x][y]){
                            ds.Union(curr, x * m + y);
                        }
                    }
                    ans = max(ans, ds.size[ds.findParent(curr)]);
                }
            }
        }

        for(int i = 0; i < n; i++){
            for(int j = 0; j < m; j++){
                if(grid[i][j] == 0){
                    map<int, int> par;
                    int curr = i * m + j;
                    int conn = 0;
                    for(int t = 0; t < 4; t++){
                        int x = i + dir[t], y = j + dir[3-t];
                        if(x >= 0 && y >= 0 && x < n && y < m && grid[x][y]){
                            int nbr = x * m + y;
                            if(par.find(ds.findParent(nbr)) == par.end()){
                                conn += ds.size[ds.findParent(nbr)];
                                par[ds.findParent(nbr)] = 1;
                            }
                        }
                    }
                    ans = max(ans, conn + 1);
                }
            }
        }

        return ans;
        
    }
};
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