<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:
        count = defaultdict(int)
        root = [[[x, y] for y in range(len(grid[0]))] for x in range(len(grid))]
        visited = [[False for y in range(len(grid[0]))] for x in range(len(grid))]
        ans = 0
        def dfs(c_coord, s_coord):
            [cx, cy] = c_coord
            [sx, sy] = s_coord
            visited[cx][cy] = True
            root[cx][cy] = s_coord
            count[(sx, sy)] += 1
            children = []
            for [dx, dy] in [[0,1], [1,0], [-1,0], [0,-1]]:
                nx = cx + dx
                ny = cy + dy
                if 0 <= nx < len(grid) and 0 <= ny < len(grid[0]):
                    if not visited[nx][ny] and grid[nx][ny] == 1:
                        dfs([nx,ny], s_coord)
                
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                if not visited[i][j] and grid[i][j] == 1:
                    dfs([i,j],[i,j])
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                if grid[i][j] == 1:
                    ans = max(ans, count[(root[i][j][0], root[i][j][1])])
                else:
                    root_set = set()
                    for [dx, dy] in [[0,1], [1,0], [-1,0], [0,-1]]:
                        nx = i + dx
                        ny = j + dy
                        if 0 <= nx < len(grid) and 0 <= ny < len(grid[0]):
                            if grid[nx][ny] == 1:
                                root_set.add((root[nx][ny][0],root[nx][ny][1]))
                    local_max = 1
                    for ele in root_set:
                        local_max += count[ele]
                    ans = max(ans, local_max)
        return ans
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