<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:
    def largestIsland(self, grid: List[List[int]]) -> int:
      color_cnt = defaultdict(int)
      grid_color = {} #(i, j): color
      
      def dfs(i, j, c):
        grid_color[i, j] = c
        color_cnt[c] += 1
        for next_i, next_j in ((i+1, j), (i-1, j), (i, j+1), (i, j-1)):
          if 0 <= next_i < m and 0 <= next_j < n and grid[next_i][next_j] and (next_i, next_j) not in grid_color:
            dfs(next_i, next_j, c)
      
      c = 0
      m, n = len(grid), len(grid[0])
      for i in range(m):
        for j in range(n):
          if grid[i][j] and (i, j) not in grid_color:
            dfs(i, j, c)
            c += 1
      
      res = max(color_cnt.values(), default=0)
      for i in range(m):
        for j in range(n):
          #flip the grid from 0 to 1
          if grid[i][j] == 0:
            colors_set = set()
            for (next_i, next_j) in ((i+1, j), (i-1, j), (i, j+1), (i, j-1)):
              if (next_i, next_j) in grid_color:
                colors_set.add(grid_color[next_i, next_j])
            res = max(res, 1 + sum(color_cnt[c] for c in colors_set))
      return res
            

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