<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)

        # Locate all zeros and initialize a sum for them
        sums = dict()
        for row, col in product(range(n), range(n)):
            if grid[row][col] == 0:
                sums[(row, col)] = 0

        if len(sums) == 0:
            return n ** 2
        
        # Find connected components
        # Assume I can change grid. Otherwise copy it.
        for row, col in product(range(n), range(n)):
            if grid[row][col] == 1:
                size, zero_neighbors = self.connected_component(grid, row, col)
                for zero_loc in zero_neighbors:
                    sums[zero_loc] += size

        for row, col in product(range(n), range(n)):
            if grid[row][col] == 2:
                grid[row][col] = 1
        
        return max(sums.values()) + 1

    def connected_component(self, grid: List[List[int]], row: int, col: int) -> Tuple[int, List[Tuple[int, int]]]:
        stack = [(row, col)]
        grid[row][col] = 2
        size = 0
        zero_neighbors = set()

        while len(stack) > 0:
            row, col = stack.pop()
            size += 1
            for row, col in self.get_neighbors(len(grid), row, col):
                if grid[row][col] == 1:
                    stack.append((row, col))
                    grid[row][col] = 2
                elif grid[row][col] == 0:
                    zero_neighbors.add((row, col))
            
        return size, zero_neighbors

    @staticmethod
    def get_neighbors(length: int, row: int, col: int) -> List[Tuple[int, int]]:
        neighbors = []
        if row > 0:
            neighbors.append((row - 1, col))
        if row < length - 1:
            neighbors.append((row + 1, col))
        if col > 0:
            neighbors.append((row, col - 1))
        if col < length - 1:
            neighbors.append((row, col + 1))
        return neighbors
        
            
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