<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 bfs(self, grid, islandStart):
        for i in range(len(grid)):
            for j in range(len(grid[i])):
                queue = deque()
                if grid[i][j] > 0 and (i, j) not in islandStart:
                    queue.append([i, j])
                    islandStart[(i, j)] = (i, j)
                size = 0
                while queue:
                    x, y = queue.popleft()
                    size += 1
                    for xdir, ydir in [[-1, 0], [1, 0], [0, -1], [0, 1]]:
                        if 0 <= x + xdir < len(grid) and 0 <= y + ydir < len(grid[i]):
                            if grid[x + xdir][y + ydir] > 0 and (x + xdir, y + ydir) not in islandStart:
                                queue.append([x + xdir, y + ydir])
                                islandStart[(x + xdir, y + ydir)] = (i, j)
                if size > 0:
                    grid[i][j] = size

                        

    def largestIsland(self, grid: List[List[int]]) -> int:
        islandStart = {}
        self.bfs(grid, islandStart)
        maxSize = 0
        for i in range(len(grid)):
            for j in range(len(grid[i])):
                if grid[i][j] > 0:
                    maxSize = max(maxSize, grid[i][j])
                    continue
                size = 1
                seen = set()
                for idir, jdir in [[-1, 0], [1, 0], [0, -1], [0, 1]]:
                    if 0 <= i + idir < len(grid) and 0 <= j + jdir < len(grid[i]):
                        if grid[i + idir][j + jdir] > 0:
                            x, y = islandStart[(i + idir, j + jdir)]
                            if (x, y) not in seen:
                                size += grid[x][y]
                                seen.add((x, y))
                maxSize = max(maxSize, size)
        return maxSize


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