<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:
        n = len(grid)
        has_zero = has_one = False
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 0:
                    has_zero = True
                if grid[r][c]:
                    has_one = True
        
        if not has_zero:
            return n**2
        
        if not has_one:
            return 1
        
        directions = [(0, 1), (0, -1), (-1, 0), (1, 0)]
        idx2island = {}
        areas = {}
        island_num = 0
        for r in range(n):
            for c in range(n):
                if grid[r][c] and (r,c) not in idx2island:
                    idx2island[(r,c)] = island_num
                    stack = [(r,c)]
                    area = 1
                    while stack:
                        node = stack.pop()
                        for dr, dc in directions:
                            newR, newC = node[0]+dr, node[1]+dc
                            if 0 <= newR < n and 0 <= newC < n and grid[newR][newC]\
                                and (newR, newC) not in idx2island:
                                idx2island[(newR, newC)] = island_num
                                area += 1
                                stack.append((newR, newC))
                    areas[island_num] = area
                    island_num += 1
        
        res = 1
        for r in range(n):
            for c in range(n):
                if not grid[r][c]:
                    nei_islands = set()
                    for dr, dc in directions:
                        newR, newC = r+dr, c+dc
                        if 0 <= newR < n and 0 <= newC < n and grid[newR][newC]:
                            nei_islands.add(idx2island[(newR, newC)])
                    local_res = 1
                    for nei in nei_islands:
                        local_res += areas[nei]
                    res = max(res, local_res)
        return res
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