<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:
    unordered_map<int,int> parent;
    int find(int x) {
      if (x != parent[x]) return find(parent[x]);
      return x;
    }
  
    void uni(int x, int y) {
      int parent_x = find(x);
      int parent_y = find(y);
      parent[parent_x] = parent_y;
    }
  
    int largestIsland(vector<vector<int>>& grid) {
      int n = grid.size();
      for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
          if (grid[i][j] == 1) parent[i*n+j] = i*n+j;
        }
      }
      for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
          if (grid[i][j] == 1) {
            if (i > 0 && grid[i-1][j] == 1) uni(i*n+j, (i-1)*n+j);
            if (i < n-1 && grid[i+1][j] == 1) uni(i*n+j, (i+1)*n+j);
            if (j > 0 && grid[i][j-1] == 1) uni(i*n+j, i*n+j-1);
            if (j < n-1 && grid[i][j+1] == 1) uni(i*n+j, i*n+j+1);
          }
        }
      }
      unordered_map<int,int> area_map;
      for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
          if (grid[i][j] == 1) {
            int root = find(i*n+j);
            area_map[root]++;
          }
        }
      }
      if (area_map.size() == 0) return 1;
      if (area_map.size() == 1)
          return area_map.begin()->second == n * n ? 
                   n * n : area_map.begin()->second + 1;
      int max_area = 0;
      for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
          if (grid[i][j] == 0) {
            set<int> s;
            if (i > 0 && grid[i-1][j] == 1) s.insert(find((i-1)*n+j));
            if (i < n-1 && grid[i+1][j] == 1) s.insert(find((i+1)*n+j));
            if (j > 0 && grid[i][j-1] == 1) s.insert(find(i*n+j-1));
            if (j < n-1 && grid[i][j+1] == 1) s.insert(find(i*n+j+1));
            int total_area = 0;
            for (int x : s) total_area += area_map[x];
            max_area = max(max_area, total_area + 1);
          }
        }
      }
      return max_area;
    }
};
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