<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>

from typing import List

class Solution:
    class DisjointSet:
        def __init__(self, n):
            self.parent = [-1] * (n * n)
            self.size = [1] * (n * n)
        
        def findParent(self, node):
            if self.parent[node] == -1:
                return node
            self.parent[node] = self.findParent(self.parent[node]) 
            return self.parent[node]
        
        def unionBySize(self, u, v):
            pu = self.findParent(u)
            pv = self.findParent(v)
            if pu != pv:
                if self.size[pu] > self.size[pv]:
                    self.size[pu] += self.size[pv]
                    self.parent[pv] = pu
                else:
                    self.size[pv] += self.size[pu]
                    self.parent[pu] = pv

    def largestIsland(self, grid: List[List[int]]) -> int:
        n = len(grid)
        ds = self.DisjointSet(n)
        dr = [-1, 0, 1, 0]
        dc = [0, 1, 0, -1]

        # Step 1: Perform union operations for all cells with value 1
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    continue
                for k in range(4):
                    nr = i + dr[k]
                    nc = j + dc[k]
                    if 0 <= nr < n and 0 <= nc < n and grid[nr][nc] == 1:
                        node = i * n + j
                        adjNode = nr * n + nc
                        ds.unionBySize(node, adjNode)
        maxi = 0
        all_one = True
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1:
                    continue
                all_one = False
                components = set()
                for k in range(4):
                    nr = i + dr[k]
                    nc = j + dc[k]
                    if 0 <= nr < n and 0 <= nc < n and grid[nr][nc] == 1:
                        newNode = nr * n + nc
                        pu = ds.findParent(newNode)
                        components.add(pu)
                compsize = 1
                for it in components:
                    compsize += ds.size[it]             
                maxi = max(maxi, compsize)
        if all_one:
            return n * n

        return maxi
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