<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)
        island_idx = 0
        islands = [[-1] * n for i in range(n)]
        island_sizes = []
        incs = [-1, 0, 1, 0, -1]
        neighbor_islands = collections.defaultdict(set)
        def dfs(i, j, idx):
            if grid[i][j] == 0:
                neighbor_islands[(i, j)].add(idx)
                return 0
            if islands[i][j] != -1: return 0
            islands[i][j] = idx
            res = 1
            for di, dj in zip(incs, incs[1:]):
                ii, jj = i + di, j + dj
                if 0 <= ii < n and 0 <= jj < n:
                    res += dfs(ii, jj, idx)
            return res
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0 or islands[i][j] != -1: continue
                size = dfs(i, j, island_idx)
                island_sizes.append(size)
                island_idx += 1
        res = max(island_sizes, default=1)
        for ij in neighbor_islands:
            size = sum(island_sizes[i] for i in neighbor_islands[ij])
            res = max(size + 1, res)
        return res


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