<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:
        cell_island_map = {}
        group_size = {}

        n = len(grid)
        visited = [[False for i in range(n)] for j in range(n)]
        # contruct island map
        group_id = 1
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1 and not visited[i][j]:
                    size = self.construct_island(grid, i, j, visited, cell_island_map, group_id)
                    group_size[group_id] = size
                    group_id += 1
        largest = 0
        
        # find largest island
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    directions = [[-1, 0], [1, 0], [0, -1], [0, 1]]
                    new_island = set()
                    curr_largest = 1
                    for direction in directions:
                        x, y = i + direction[0], j + direction[1]
                        if x >=0 and x < n and y >= 0 and y < n and grid[x][y] == 1:
                            if cell_island_map[x*n + y] not in new_island:
                                new_island.add(cell_island_map[x*n + y])
                                curr_largest += group_size[cell_island_map[x*n + y]]
                    largest = max(largest, curr_largest)
        return n*n if largest == 0 else largest

    def construct_island(self, grid, i, j, visited, cell_island_map, group_id):
        n = len(grid)
        if i < 0 or i >=n or j < 0 or j >= n or grid[i][j] == 0 or visited[i][j]:
            return 0
        p = i*n + j
        cell_island_map[p] = group_id
        visited[i][j] = True
        directions = [[-1, 0], [1, 0], [0, -1], [0, 1]]
        size = 1
        for direction in directions:
            x = i + direction[0]
            y = j + direction[1]
            size += self.construct_island(grid, x, y, visited, cell_island_map, group_id)
        return size

        
        
        
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