<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:
        # step 1 bfs find all the islands
        if not grid or not grid[0]:
            return 0
        n = len(grid)
        to_visit = {(i, j) for i in range(n) for j in range(n)}
        islands = {} # islands each cell belongs to
        size = {} # size of each islands

        while to_visit:
            q = deque()
            i, j = to_visit.pop()
            if grid[i][j] == 1:
                curr_size = 0

                q.append((i, j))
                while q:
                    x, y = q.popleft()
                    curr_size += 1
                    islands[(x, y)] = (i, j)
                    for xx, yy in [(x-1, y), (x+1, y), (x, y-1), (x, y+1)]:
                        if -1 < xx < n and -1 < yy < n and (xx, yy) in to_visit:
                            to_visit.remove((xx, yy))
                            if grid[xx][yy] == 1:
                                q.append((xx, yy))
                # we finished one island (i, j)
                size[(i, j)] = curr_size
        ans = max(size.values()) if size else 0
        # step 2, flip 0 to find largest
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    neighbors = set() 
                    for x, y in [(i-1, j), (i+1, j), (i, j-1), (i, j+1)]:
                        if (x, y) in islands:
                            neighbors.add(islands[(x, y)])
                    ans = max(ans, sum(size[idx] for idx in neighbors)+1)
        
        return ans
            



        
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