<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:
        island_index = 0
        island_area = {}
        coordinate_index = {}
        m,n = len(grid), len(grid[0])
        
        def mark_island(r,c,index):
            if 0<=r<m and 0<=c<n and (r,c) not in coordinate_index and  grid[r][c] == 1:
                coordinate_index[(r,c)] = index
                return 1 + mark_island(r-1,c,index) + mark_island(r+1,c,index)+ mark_island(r,c-1,index)+ mark_island(r,c+1,index)
            return 0
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 1 and (i,j) not in coordinate_index:
                    area = mark_island(i,j, island_index)
                    island_area[island_index] = area
                    island_index += 1
        if not island_area:
            return 1
        max_area = max(island_area.values())
        
        # print(island_area, coordinate_index)
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 0:
                    visited_index = set()
                    if (i-1, j) in coordinate_index:
                        visited_index.add(coordinate_index[(i-1, j)])
                    if (i+1, j) in coordinate_index:
                        visited_index.add(coordinate_index[(i+1, j)])
                    if (i, j-1) in coordinate_index:
                        visited_index.add(coordinate_index[(i, j-1)])
                    if (i, j+1) in coordinate_index:
                        visited_index.add(coordinate_index[(i, j+1)])
                    area = 1
                    while len(visited_index):
                        area += island_area[visited_index.pop()]
                    max_area=max(area, max_area)
        return max_area
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