<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:
        island_id_map = {}
        island_id_size_map = {}
        m, n = len(grid), len(grid[0])
        stack, visited, zeros = [], set(), []
        island_id = island_size = 0
        largest_island = 0
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 1 and (i, j) not in visited:
                    stack.append((i, j))
                    visited.add((i, j))
                    island_id += 1
                    island_size = 1
                elif grid[i][j] == 0:
                    zeros.append((i, j))
                while stack:
                    a, b = stack.pop()
                    island_id_map[(a, b)] = island_id
                    for u, v in (0, 1), (1, 0), (-1, 0), (0, -1):
                        x, y = a + u, b + v
                        if 0 <= x < m and 0 <= y < n and grid[x][y] == 1 and (x, y) not in visited:
                            stack.append((x, y))
                            visited.add((x, y))
                            island_size += 1
                island_id_size_map[island_id] = island_size
                largest_island = max(largest_island, island_size)

        visited_islands = set()
        for a, b in zeros:
            current_island = 1
            for u, v in (0, 1), (1, 0), (-1, 0), (0, -1):
                x, y = a + u, b + v
                if 0 <= x < m and 0 <= y < n and (x, y) in island_id_map and island_id_map[(x, y)] not in visited_islands:
                    current_island += island_id_size_map[island_id_map[(x, y)]]
                    visited_islands.add(island_id_map[(x, y)])
            largest_island = max(largest_island, current_island)
            visited_islands = set()
        
        return largest_island


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