<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)]
        size = [1 for i in range(n*n)]
        directions = [(0,1),(0,-1),(1,0),(-1,0)]

        def isValid(x, y):
            if 0 <= x < n and 0 <= y < n:
                return True
            return False

        def find(node):
            if parent[node] != node:
                parent[node] = find(parent[node])
            return parent[node]

        def union(x, y): # By Size
            root_x, root_y = find(x), find(y)
            if root_x != root_y:
                if size[root_y] > size[root_x]:
                    parent[root_x] = root_y
                    size[root_y] += size[root_x]
                else:
                    parent[root_y] = root_x
                    size[root_x] += size[root_y]

        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    continue
                for dx, dy in directions:
                    a, b = i + dx, j + dy
                    if isValid(a, b) and grid[a][b] == 1:
                        union((n * i) + j, (n * a) + b)
        
        max_area = 0
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1:
                    max_area = max(max_area, size[(n * i) + j])
                else:
                    base = set()
                    for dx, dy in directions:
                        a, b = i + dx, j + dy
                        if isValid(a, b) and grid[a][b] == 1:
                            base.add(find((n * a) + b))
                    val = 0
                    for k in base:
                        val += size[k]
                    max_area = max(val + 1, max_area)
        return max_area

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