<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:
        ROWS, COLS = len(grid), len(grid[0])
        n = ROWS * COLS
        root = [i for i in range(n)]
        rank = [1] * n

        def find(x):
            if x != root[x]:
                root[x] = find(root[x])
            return root[x]

        def union(x, y):
            rootX = find(x)
            rootY = find(y)
            if rootX != rootY:
                if rank[rootX] > rank[rootY]:
                    root[rootY] = rootX
                elif rank[rootX] < rank[rootY]:
                    root[rootX] = rootY
                else:
                    root[rootX] = rootY
                    rank[rootY]+=1
        dirs = [(0, 1), (0, -1), (1, 0), (-1, 0)]

        for r in range(ROWS):
            for c in range(COLS):
                if grid[r][c] == 1:
                    index = r * COLS + c
                    for dr, dc in dirs:
                        nr, nc = r+dr, c+dc
                        if nr < 0 or nr >= ROWS or nc < 0 or nc >= COLS or grid[nr][nc]==0:
                            continue
                        newIndex = nr*COLS+nc
                        union(index, newIndex)
        size = defaultdict(int)
        for i in range(n):
            size[find(i)]+=1
        
        res = 0
        for r in range(ROWS):
            for c in range(COLS):
                if grid[r][c] == 1:
                    res = max(res, size[find(r*COLS+c)])
                else:
                    cand = 1
                    seen = set()
                    for dr, dc in dirs:
                        nr, nc = r+dr, c+dc
                        if nr < 0 or nr >= ROWS or nc < 0 or nc >= COLS or grid[nr][nc] == 0:
                            continue
                        newIndex = find(nr*COLS+nc)
                        if newIndex not in seen:
                            seen.add(newIndex)
                            cand+=size[newIndex]
                    res = max(cand, res)
        return res
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