<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:
    M, N = len(grid), len(grid[0])

    def labelIslandAndGetArea(row, col, label):
      q = deque([(row, col)])
      grid[row][col] = label
      area = 1

      while q:
        curRow, curCol = q.popleft()
        for dr, dc in [(-1, 0), (0, -1), (1, 0), (0, 1)]:
          nr, nc = curRow + dr, curCol + dc
          if 0 <= nr < M and 0 <= nc < N and grid[nr][nc] == 1:
            grid[nr][nc] = label
            q.append((nr, nc))
            area += 1
      
      return area
    
    areas = dict()
    label = 2
    seenZero = False
    maxArea = 0
    for row in range(M):
      for col in range(N):
        if grid[row][col] == 1:
          area = labelIslandAndGetArea(row, col, label)
          areas[label] = area
          label += 1
          maxArea = max(maxArea, area)
        else:
          seenZero = True
    
    if not seenZero:
      return maxArea
    
    for row in range(M):
      for col in range(N):
        if grid[row][col] == 0:
          myArea = 1 # flipping this 0 to 1 gets an area of 1
          connectedAreas = set()
          for dr, dc in [(-1, 0), (0, -1), (1, 0), (0, 1)]:
            nr, nc = row + dr, col + dc
            if 0 <= nr < M and 0 <= nc < N and grid[nr][nc] > 1:
              connectedAreas.add(grid[nr][nc])
          for areaLabel in connectedAreas:
            myArea += areas[areaLabel]
          maxArea = max(maxArea, myArea)
    
    return maxArea
            



        
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