<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:
        root_counts = {}
        zeros = []
        index = 1
        rows = len(grid)
        cols = len(grid[0])
        def _valid_neighbors(i, j):
            neighbors = [(i+1, j), (i-1, j), (i, j-1), (i, j+1)]
            returns = []
            for ii, jj in neighbors:
                if ii < 0 or ii >= rows or jj < 0 or jj >= cols:
                    continue
                returns.append((ii, jj))
            return returns

        for i in range(rows):
            for j in range(cols):
                if grid[i][j] != 1:
                    if grid[i][j] == 0:
                        zeros.append((i, j))
                    continue
                index += 1
                stack = [(i, j)]
                size = 0
                while stack:
                    ii, jj = stack.pop()
                    if grid[ii][jj] != 1:
                        continue
                    grid[ii][jj] = index
                    size += 1
                    #print(f"{ii}, {jj}. {size}, {grid[ii][jj]}")
                    for c_i, c_j in _valid_neighbors(ii, jj):
                        if grid[c_i][c_j] == 1:
                            stack.append((c_i, c_j))
                root_counts[index] = size
        max_island = max(root_counts.values()) if root_counts else 0
        #print(root_counts)
        for z_i, z_j in zeros:
            max_size = 0
            seen = set()
            for neighbor in _valid_neighbors(z_i, z_j):
                index = grid[neighbor[0]][neighbor[1]]
                if index > 0 and index not in seen:
                    max_size += root_counts[index]
                    seen.add(index)
            max_island = max(max_island, max_size+1) 
        return max_island 

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