<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:
        
        n = len(grid)

        def dfs(row, col, island_id):
            if(not(0 <= row < n and 0 <= col < n) or grid[row][col] != 1):
                return 0

            grid[row][col] = island_id
            size = 1

            for nrow, ncol in [[0,-1],[-1,0],[1,0],[0,1]]:
                size += dfs(nrow+row, ncol+col, island_id)

            return size

        island_sizes = {}
        island_id = 2

        for row in range(n):
            for col in range(n):
                if(grid[row][col] == 1):
                    island_sizes[island_id] = dfs(row, col, island_id)
                    island_id += 1


        max_size = max(island_sizes.values()) if island_sizes else 0
        for row in range(n):
            for col in range(n):
                if(grid[row][col] == 0):
                    neighbor_islands = set()
                    for nrow, ncol in [[0,-1],[-1,0],[1,0],[0,1]]:
                        nrow += row
                        ncol += col

                        if(0 <= nrow < n and 0 <= ncol < n and grid[nrow][ncol] > 1):
                            neighbor_islands.add(grid[nrow][ncol])

                    size = 1 + sum(island_sizes[island] for island in neighbor_islands)

                    max_size = max(max_size, size)

        return max_size
        

    


      
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