<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>

def dfs(grid, root_cell, visited_set, island):
    visited_set.add(root_cell)
    island.add(root_cell)
    x = root_cell[0]
    y = root_cell[1]
    n = len(grid)

    for (i,j) in [(0,1),(0,-1), (-1,0), (1,0)]:
        if x+i >=0 and x+i < n and y+j>=0 and y+j < n and (x+i,y+j) not in visited_set and grid[x+i][y+j]:
            dfs(grid, (x+i,y+j), visited_set, island)

class Solution:
    def largestIsland(self, grid: List[List[int]]) -> int:
        
        if sum([sum(x) for x in grid]) == 0:
            return 1

        islands = []
        n = len(grid)
        visited_set = set()
        islands = []
        for i in range(n):
            for j in range(n):
                if (i,j) not in visited_set and grid[i][j]:
                    island = set()
                    dfs(grid, (i,j), visited_set, island)
                    islands.append(island)

        
        disjoint_sets = {}
        island_id = 0
        island_areas = {}
        for isl in islands:
            for cell in isl:
                disjoint_sets[cell] = island_id
            island_areas[island_id] = len(isl)
            island_id+=1


        max_size = max(island_areas.values())
        for x in range(n):
            for y in range(n):
                if not(grid[x][y]):
                    new_size = 1
                    visited_island = set()
                    for (i,j) in [(0,1),(0,-1),(1,0), (-1,0)]:
                        island_id = disjoint_sets.get((x+i,y+j))
                        if island_id!=None and (island_id not in visited_island):
                            new_size += island_areas.get(island_id,0) 
                            visited_island.add(island_id)
                    max_size = max(max_size, new_size)

        return max_size

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