<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)
        DIRECTIONS = [(-1,0),(0,-1),(0,1),(1,0)]

        address = {}
        def bfs(row, column, island_id):
            queue = deque([(row, column, island_id)])
            visited.add((row, column))
            area = 1

            while queue:
                row, column, island_id = queue.popleft()
                address[(row, column)] = island_id
                for direction in DIRECTIONS:
                    r, c = row+direction[0], column+direction[1]
                    if 0<=r<N and 0<=c<N and grid[r][c]==1 and (r,c) not in visited:
                        queue.append((r,c,island_id))
                        visited.add((r,c))
                        area += 1
            return area
        
        visited = set()
        area = {}
        island_id = 0
        for row in range(N):
            for col in range(N):
                if grid[row][col]==1 and (row, col) not in visited:
                    area[island_id] = bfs(row, col, island_id)
                    island_id += 1
        
        if len(address.keys()) == N**2:
            return N**2
        
        largest_area = 1
        for row in range(N):
            for col in range(N):
                if grid[row][col]==0:
                    neighbors = set()
                    large_area = 1
                    for direction in DIRECTIONS:
                        r, c = row + direction[0], col + direction[1]
                        if 0<=r<N and 0<=c<N and grid[r][c]==1 and address[(r,c)] not in neighbors:
                            neighbors.add(address[(r,c)])
                            large_area += area[address[(r,c)]]
                    largest_area = max(largest_area, large_area)
        return largest_area
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