<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:
        group = 0
        sizes = {}
        n = len(grid)
        visited = set()
        groups = [[None for _ in range(n)] for _ in range(n)]
        nbrs = [(0, -1), (-1, 0), (1, 0), (0, 1)]
        size =  0

        def dfs(i, j, group, size):   # modifies groups
            visited.add((i, j))
            size += 1
            groups[i][j] = group
            for nbr in nbrs:
                x = i + nbr[0]
                y = j + nbr[1]
                if x < 0 or x >=n:  continue
                if y < 0 or y >= n: continue
                if (x, y) in visited or grid[x][y] == 0:   continue
                size = dfs(x, y, group, size)
            return size

        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1 and (i,j) not in visited:
                    sizes[group] = dfs(i, j, group, 0)
                    group += 1
        
        # print(groups)
        # print(sizes)

        result = 0
        if sizes:   result = max(sizes.values())
        if not sizes:   return 1
        # size by expansion
        for i in range(n):
            for j in range(n):
                if grid[i][j] != 0: continue
                nbr_groups = set()
                # print('i & j:', i, j)
                for nbr in nbrs:
                    x, y = i + nbr[0], j + nbr[1]
                    if x < 0 or y < 0 or x >= n or y >= n:  continue
                    # print('nbr:', x, y)
                    if groups[x][y] is not None:
                        # print('nbr is ', x, ' and ', y, 'with group -', groups[x][y])
                        nbr_groups.add(groups[x][y])
                # print(i, j, nbr_groups)
                joined_size = 1 + sum([sizes[grp] for grp in nbr_groups])
                result = max(result, joined_size)

        return result
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