<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:
        # use dp to store the size of island 
        # then the islands with 0 digits, find the size of that island
        n = len(grid)
        # parent of given node
        parents = [i for i in range(n*n)]
        # size of the island
        set_sizes = [1] * (n*n)

        def find_root(el):
            # path compression is done here
            if parents[el] != el:
                parents[el] = find_root(parents[el])
            return parents[el]
        
        def union(x, y):
            px = find_root(x)
            py = find_root(y)
            if px != py:
                if set_sizes[px] < set_sizes[py]:
                    parents[px] = py
                    set_sizes[py] += set_sizes[px]
                else:
                    parents[py] = px
                    set_sizes[px] += set_sizes[py]
        
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 1:
                    # union with neighbors left and above
                    dirs = [[-1,0], [0, -1]]
                    for dr, dc in dirs:
                        nr = r + dr
                        nc = c + dc
                        if 0 <= nr < n and 0 <= nc < n and grid[nr][nc] == 1:
                            union(r * n + c, nr * n + nc)
        print(set_sizes)
        print(parents)
        max_island_size = max(set_sizes)
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 0:
                    visited_roots = set()
                    temp_size = 1
                    dirs = [[0,-1], [1,0], [-1, 0], [0, 1]]
                    for dr, dc in dirs:
                        nr = r + dr
                        nc = c + dc
                        if 0 <= nr < n and 0 <= nc < n and grid[nr][nc] == 1:
                            root = find_root(nr * n + nc)
                            if root not in visited_roots:
                                visited_roots.add(root)
                                temp_size += set_sizes[root]
                    # update if we have found a larger island
                    max_island_size = max(max_island_size, temp_size)
        return max_island_size
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