<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 DisjointSet:
    def __init__(self, n) -> None:
        self.parent = [i for i in range(n)]
        self.size = [1 for i in range(n)]

    def findParent(self, node):
        res = node
        while res != self.parent[res]:
            self.parent[res] = self.parent[self.parent[res]]
            res = self.parent[res]
        return res

    def unionBySize(self, n1, n2):
        p1 = self.findParent(n1)
        p2 = self.findParent(n2)
        if p1 == p2:
            return
        if self.size[p1] < self.size[p2]:
            self.size[p2] += self.size[p1]
            self.parent[p1] = self.parent[p2]
        else:
            self.size[p1] += self.size[p2]
            self.parent[p2] = self.parent[p1]

class Solution:
    def largestIsland(self, grid: List[List[int]]) -> int:
        n = len(grid)
        ds = DisjointSet(n*n)
        directions = [[-1, 0], [0, 1], [1, 0], [0, -1]]
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 0:
                    continue
                for dr, dc in directions:
                    adjR = r+dr
                    adjC = c+dc
                    if min(adjR, adjC) >= 0 and max(adjR, adjC) < n and grid[adjR][adjC] == 1:
                        node = r*n + c
                        adjNode = adjR*n + adjC
                        ds.unionBySize(node, adjNode)
        maxIslandSize = 0
        sizeTotal = 0
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 1:
                    continue
                islandSet = set()
                sizeTotal = 0
                for dr, dc in directions:
                    adjR = r+dr
                    adjC = c+dc
                    if min(adjR, adjC) >= 0 and max(adjR, adjC) < n and grid[adjR][adjC] == 1:
                        islandSet.add(ds.findParent(adjR*n+adjC))
                for island in islandSet:
                    sizeTotal += ds.size[island]
                maxIslandSize = max(maxIslandSize, sizeTotal+1)

        for node in range(n*n):
            maxIslandSize = max(maxIslandSize, ds.size[node])
        return maxIslandSize
        
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