<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])

        idx = lambda i, j: m * i + j    
        nbh = lambda i, j: [
            (i1, j1) for i1, j1 in [(i+1, j), (i-1, j), (i, j+1), (i, j-1)] 
            if 0 <= i1 < m and 0 <= j1 < n
        ]

        parent = [[None] * m for _ in range(n)]
        def dfs(i, j, p):
            if i < 0 or i >= m or j < 0 or j >= n or grid[i][j] == 0:
                return 0

            size = 1
            parent[i][j] = p
            for i1, j1 in nbh(i, j):
                if parent[i1][j1] is None:
                    size += dfs(i1, j1, p)
            return size

        island = dict()
        ans = 0
        for i in range(m): 
            for j in range(n):
                if grid[i][j] == 1 and parent[i][j] is None:
                    k = idx(i, j)
                    island[k] = dfs(i, j, k)
                    ans = max(ans, island[k])

        # print(parent)
        # print(island)

        for i in range(m):
            for j in range(n):
                if grid[i][j] == 0:
                    neibor_island = {parent[i1][j1] for i1, j1 in nbh(i, j) if parent[i1][j1] is not None}
                    tmp = sum(island[k] for k in neibor_island)
                    ans = max(ans, 1 + tmp)
                    # print(f'after {(i, j)}, {ans=}, {tmp=}, {neibor_island=}')

        return ans
31093