<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>

def find_parent(node, parents):
    if parents[node] == node:
        return node
    
    parents[node] = find_parent(parents[node], parents)
    return parents[node]

def union(u, v, parents, ranks, sizes):
    u = find_parent(u, parents)
    v = find_parent(v, parents)

    if u == v: 
        return

    if ranks[u] > ranks[v]:
        parents[v] = u
        sizes[u] += sizes[v]
    elif ranks[v] > ranks[u]:
        parents[u] = v
        sizes[v] += sizes[u]
    else:
        parents[v] = u
        ranks[u] += 1
        sizes[u] += sizes[v]


class Solution:
    def largestIsland(self, grid: List[List[int]]) -> int:
        n = len(grid)
        nxn = n * n

        # intialize union set
        parents = [i for i in range(nxn)]
        ranks = [0] * nxn 
        sizes = [1] * nxn

        # merge to island if edges exists

        for row in range(n):
            for col in range(n):
                if grid[row][col] == 0:
                    continue
                u = row * n + col
                
                if 0 <= row + 1 < n and grid[row + 1][col] == 1:
                    v = u + n
                    union(u, v, parents, ranks, sizes)
                
                if 0 <= col + 1 < n and grid[row][col+1] == 1:
                    v = u + 1
                    union(u, v, parents, ranks, sizes)
                
        max_island_size = max(sizes)

        for row in range(n):
            for col in range(n):
                if grid[row][col] == 1:
                    continue
                
                u = row * n + col
                
                island_set = set()

                if 0 <= row - 1 < n and grid[row - 1][col] == 1:
                    v = u - n
                    island_set.add(find_parent(v, parents))

                if 0 <= row + 1 < n and grid[row + 1][col] == 1:
                    v = u + n
                    island_set.add(find_parent(v, parents))
                
                if 0 <= col - 1 < n and grid[row][col-1] == 1:
                    v = u - 1
                    island_set.add(find_parent(v, parents))

                if 0 <= col + 1 < n and grid[row][col+1] == 1:
                    v = u + 1
                    island_set.add(find_parent(v, parents))
                
                island_size = 1
                for island in island_set:
                    island_size += sizes[island]
                
                max_island_size = max(max_island_size, island_size)

        return max_island_size
                

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