<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 = {}
        island_num = 0
        def recurse(i, j):
            if i < 0 or i >= len(grid) or j < 0 or j >= len(grid[0]) or (i, j) in visited or grid[i][j] == 0:
                return 0
            
            nonlocal island_num
            visited[(i, j)] = island_num

            return 1 + recurse(i, j-1) + recurse(i, j+1) + recurse(i-1, j) + recurse(i+1, j)

        num_islands = 0
        island_num_area_map = {}
        cur_max = 0
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                if (i, j) not in visited and grid[i][j] == 1:
                    island_num_area_map[island_num] = recurse(i, j)
                    cur_max = max(island_num_area_map[island_num], cur_max)
                    island_num += 1
                    num_islands += 1
        
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                if grid[i][j] == 0:
                    island_set = set()
                    for pos in [(0, -1), (0, 1), (-1, 0), (1, 0)]:
                        new_i = i+pos[0]
                        new_j = j+pos[1]

                        if new_i < 0 or new_i >= len(grid) or new_j < 0 or new_j >= len(grid[0]) or grid[new_i][new_j] == 0:
                            continue
                        
                        island_set.add(visited[(new_i, new_j)])
                    
                    total = 1
                    for island in island_set:
                        total += island_num_area_map[island]
                    cur_max = max(total, cur_max)
            
        return cur_max

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