<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)
        parent=[i for i in range(n*n)]
        rank = [1 for i in range(n*n)]
        sizes = [1 for i in range(n*n)]
        maxsize = 0

        def find(x):
            nonlocal maxsize
            maxsize = max(1, maxsize)
            if x != parent[x]:
                parent[x] = find(parent[x])
            return parent[x]
        
        def union(x,y):
            rx,ry = find(x), find(y)
            nonlocal maxsize
            if rx != ry:
                if rank[rx] < rank[ry]:
                    parent[rx] =  ry
                    sizes[ry] += sizes[rx]
                    maxsize = max(maxsize, sizes[ry])
                else:
                    parent[ry] = rx
                    sizes[rx] += sizes[ry]
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    maxsize = max(maxsize, sizes[rx])

            return

        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1:
                    find(i*n + j)
                    for di,dj in [(0,1),(1,0),(0,-1),(-1,0)]:
                        ni,nj =i+di, j+dj
                        if 0 <= ni < n and 0 <= nj < n and grid[ni][nj] == 1:
                            union(i*n + j, ni*n + nj)

        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    size = 0
                    pars = set()
                    for di,dj in [(0,1),(1,0),(0,-1),(-1,0)]:
                        ni,nj =i+di, j+dj
                        if 0 <= ni < n and 0 <= nj < n and grid[ni][nj] == 1:
                            pars.add(find(ni*n + nj))
                    for par in pars:
                        size += sizes[par]
                    maxsize = max(size + 1, maxsize)
        return maxsize




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