<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.father = {}
        self.size_of_set = {}

    def add(self, x):
        if x in self.father:
            return
        self.father[x] = None
        self.size_of_set[x] = 1

    def find(self, x):
        root = x
        while self.father[root] is not None:
            root = self.father[root]
        while x != root:
            org_f = self.father[x]
            self.father[x] = root
            x = org_f
        return root

    def union(self, x, y):
        root_x, root_y = self.find(x), self.find(y)
        if root_x != root_y:
            self.father[root_x] = root_y
            self.size_of_set[root_y] += self.size_of_set[root_x]
            self.size_of_set.pop(root_x)
        return 


class Solution:
    def largestIsland(self, grid: List[List[int]]) -> int:
        
        ## first use union find to map out all islands
        ## enumerate over the grid to find the result
        ## time complexity O(n^2), space complexity O(n^2)

        if not grid or not grid[0]:
            return 0

        n = len(grid)
        directions = [(-1, 0), (0, 1), (1, 0), (0, -1)]
        uf = UnionFind()

        for x in range(n):
            for y in range(n):
                if grid[x][y] == 0:
                    continue
                uf.add((x, y))

                for dx, dy in directions:
                    nx, ny = x + dx, y + dy
                    if not (0 <= nx < n and 0 <= ny < n):
                        continue
                    if grid[nx][ny] == 0:
                        continue
                    
                    uf.add((nx, ny))
                    uf.union((nx, ny), (x, y))

        res = max(uf.size_of_set.values()) if uf.size_of_set else 0
        if res == 0:
            return 1

        for x in range(n):
            for y in range(n):
                if grid[x][y] == 1:
                    continue

                root_ls = set()
                area = 1
                for dx, dy in directions:
                    nx, ny = x + dx, y + dy
                    if not (0 <= nx < n and  0 <= ny < n):
                        continue
                    if grid[nx][ny] == 0:
                        continue
                    root = uf.find((nx, ny))
                    if root not in root_ls:
                        area += uf.size_of_set[root]
                        root_ls.add(root)

                res = max(res, area)

        return res
44631