<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:
        
        visited = set()#bfs
        island_to_id = {}#map island to id
        island_id_to_area = {}#map island_id to area
        island_id = 0
        n = len(grid)
        moves = [(-1, 0), (1, 0), (0, 1), (0, -1)]
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 1 and (r,c) not in visited:
                    visited.add((r,c))
                    stack = [(r,c)]
                    island_id_to_area[island_id] = 0
                    while stack:
                        x, y = stack.pop()
                        island_to_id[(x,y)] = island_id
                        island_id_to_area[island_id] += 1
                        for dx, dy in moves:
                            new_x = x + dx
                            new_y = y + dy
                            if 0 <= new_x < n and 0 <= new_y < n:
                                if grid[new_x][new_y] == 1 and (new_x, new_y) not in visited:
                                    stack.append((new_x, new_y)) 
                                    visited.add((new_x, new_y))
                    island_id += 1
                else:
                    pass
  
        if island_id_to_area:
            ans = max([area for _, area in island_id_to_area.items()])
        else:
            ans = 0
            
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 0:
                    unique_island = set()
                    for dr, dc in moves:
                        new_r = r + dr
                        new_c = c + dc       
                        if (new_r, new_c) in island_to_id:
                            unique_island.add(island_to_id[(new_r,new_c)])
                    area = 1
                    for uni_id in unique_island:
                        area += island_id_to_area[uni_id]
                    ans = max(ans, area)
        return ans
                    
                    
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