<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)
        c_id_to_size = dict() # c_id -> c size
        point_to_c = dict() # (i,j) -> c_id

        def dfs(point: tuple[int], c_id: int) -> int:
            point_to_c[point] = c_id
            subtree_size = 1
            for d_x,d_y in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
                n_x = point[0] + d_x
                n_y = point[1] + d_y
                if 0 <= n_x < n and 0 <= n_y < n and grid[n_x][n_y] == 1 and (n_x,n_y) not in point_to_c:
                    subtree_size += dfs((n_x,n_y), c_id)
            return subtree_size

        c_id = 0
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1 and not (i,j) in point_to_c:
                    c_id_to_size[c_id] = dfs((i,j), c_id)
                    c_id += 1
        
        #print(f"{point_to_c}")
        max_island_size = max(c_id_to_size.values()) if c_id_to_size else 0
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    neighboring_islands = set()
                    for d_x,d_y in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
                        n_x = i + d_x
                        n_y = j + d_y
                        if 0 <= n_x < n and 0 <= n_y < n and (n_x,n_y) in point_to_c:
                            neighboring_islands.add(point_to_c[(n_x,n_y)])
                    #print(f"i: {i}, j: {j}, neighboring islands: {neighboring_islands}")
                    island_size = 1
                    for neighboring_island in neighboring_islands:
                        island_size += c_id_to_size[neighboring_island]
                    max_island_size = max(max_island_size, island_size)

        return max_island_size


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