<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)
        is_valid = lambda x, y: (0 <= x < N) and (0 <= y < N)

        origin_grid = [[None] * N for _ in range(N)]
        origin_area = dict()

        def dfs(r, c, origin=None):
            if not is_valid(r, c):
                return 0
            
            if grid[r][c] == 0:
                return 0

            area = 1
            origin_grid[r][c] = origin or (r, c)
            for dr, dc in [(1, 0), (-1, 0), (0, 1), (0, -1)]:
                nr, nc = r + dr, c + dc
                if is_valid(nr, nc) and origin_grid[nr][nc] is None:
                    area +=  dfs(nr, nc, origin or (r, c))
            if origin is None: origin_area[(r, c)] = area
            return area
                
        for i in range(N):
            for j in range(N):
                if grid[i][j] and origin_grid[i][j] is None:
                    dfs(i, j)
        
        ans = max(origin_area.values() or [0])
        for i in range(N):
            for j in range(N):
                if grid[i][j] == 1:
                    continue

                total, added = 1, set()
                for di, dj in [(1, 0), (-1, 0), (0, 1), (0, -1)]:
                    ni, nj = i + di, j + dj
                    if (not is_valid(ni, nj)) or grid[ni][nj] == 0:
                        continue

                    origin = origin_grid[ni][nj]
                    if origin not in added:
                        total += origin_area[origin]
                        added.add(origin)
                ans = max(ans, total)

        return ans

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