<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)
        pos_island = {}
        idx_size = defaultdict(int)
        def searchIsland(row, col, grid, island_idx):
            nonlocal N, pos_island
            pos_island[(row, col)] = island_idx
            grid[row][col] = 2
            idx_size[island_idx] += 1
            for dr, dc in [(0, 1), (1, 0), (0, -1), (-1, 0)]:
                new_row = row + dr
                new_col = col + dc
                if new_row >= 0 and new_row < N and new_col >= 0 and new_col < N and grid[new_row][new_col] == 1:
                    searchIsland(new_row, new_col, grid, island_idx)
            # grid[row][col] = 1
            # return size
        
        empty_pos = []
        island_idx = 0
        for row in range(N):
            for col in range(N):
                if grid[row][col] == 1 and ((row, col) not in pos_island or pos_island[(row, col)] not in idx_size):
                    searchIsland(row, col, grid, island_idx)
                    island_idx += 1
                elif grid[row][col] == 0:
                    empty_pos.append((row, col))

        if not empty_pos:
            return N * N
        res = 1
        for row, col in empty_pos:
            island_idx_set = set()
            island_size = 1
            for dr, dc in [(0, 1), (1, 0), (0, -1), (-1, 0)]:
                new_row = row + dr
                new_col = col + dc
                if new_row >= 0 and new_row < N and new_col >= 0 and new_col < N and grid[new_row][new_col] != 0 and pos_island[(new_row, new_col)] not in island_idx_set:
                    # print(new_row, new_col)
                    island_idx_set.add(pos_island[(new_row, new_col)])
                    island_size += idx_size[pos_island[(new_row, new_col)]]
            res = max(res, island_size)
        return res

64581