<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 UnionFind:
    def __init__(self):
        self.parent = dict()
        self.size = dict()
    
    def add(self, node):
        self.parent[node] = node
        self.size[node] = 1
    
    def find(self, node):
        while self.parent[node]!=node:
            self.parent[node] = self.parent[self.parent[node]]
            node = self.parent[node]
        return node
    
    def union(self, node1, node2):
        r1, r2 = self.find(node1), self.find(node2)
        if r1 == r2:
            return
        self.parent[r2] = r1
        self.size[r1] += self.size[r2]
        self.size.pop(r2)

    def getSize(self, node):
        return self.size.get(self.find(node), 0)
    
    def getMaxSize(self):
        if len(self.size) == 0:
            return 0
        return max(self.size.values())

directions1 = [(-1, 0), (0, -1)]
directions2 = [(-1, 0), (1, 0), (0, -1), (0, 1)]
class Solution:
    def largestIsland(self, grid: List[List[int]]) -> int:
        uf = UnionFind()
        n = len(grid)
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    continue
                p = (i, j)
                uf.add(p)
                for d in directions1:
                    ni, nj = i+d[0], j+d[1]
                    if 0<=ni<n and 0<=nj<n and grid[ni][nj]==1:
                        uf.union(p, (ni, nj))
        r = uf.getMaxSize()
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1:
                    continue
                islands = set()
                for d in directions2:
                    ni, nj = i+d[0], j+d[1]
                    if 0<=ni<n and 0<=nj<n and grid[ni][nj]==1:
                        islands.add(uf.find((ni, nj)))
                r = max(r, 1+sum(map(lambda x:uf.getSize(x), islands)))
        return r
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