<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:
        # brute force: change 1 grid item, then do DFS O(n^2)
        # do union find O(n^2) and then see the resulting size everytime you
        # change 1 element O(n^2)
        n = len(grid)
        island_sizes = [1] * (n * n)
        roots = [i for i in range(n*n)]


        def find_root(x):
            if roots[x] != x:
                roots[x] = find_root(roots[x])  # Path compression
            return roots[x]

        def get_union(x, y): #8, 7
            px = find_root(x) #px=8, island_size=1
            py = find_root(y) #py=0, island_size=7
            if px != py:
                if island_sizes[px] < island_sizes[py]:
                    island_sizes[py] += island_sizes[px] # island_
                    roots[px] = py
                else: #island_sizes[0] += island_sizes[3]
                    island_sizes[px] += island_sizes[py]
                    roots[py] = px # roots[2] = 0
        # [1,1,1]
        # [1,0,1*]
        # [1,1,1]
        # roots =      [0,0,0,0,4,0,0,0,0]
        # island_sizes=[8,1,1,1,1,1,1,1,1]

        for r in range(n):
            for c in range(n):
                # check if above and left are neighbors
                if grid[r][c] == 1:
                    dirs = [[-1, 0], [0, -1]]
                    for dr, dc in dirs:
                        nr = r + dr
                        nc = c + dc
                        if 0 <= nr < n and 0 <= nc < n and grid[nr][nc] == 1:
                            get_union(r * n + c, nr * n + nc) # (8,7)
        
        max_island_size = max(island_sizes)
        # now check if island changes after you apply grid
        for r in range(n):
            for c in range(n):
                if grid[r][c] == 0: # 1,1
                    # change r,c to 1 and see what size is 
                    visited_roots = set() # 0
                    cur_size = 1
                    dirs = [[0,1],[1,0],[-1, 0], [0,-1]]
                    for dr, dc in dirs:
                        nr = r + dr # 1
                        nc = c + dc # 2
                        if 0 <= nr < n and 0 <= nc < n and grid[nr][nc] == 1:
                            root = find_root(nr * n + nc) # find_root(5)=0
                            if root not in visited_roots:
                                visited_roots.add(root)
                                cur_size += island_sizes[root] # cur_size = 1 + 8
                    max_island_size = max(max_island_size, cur_size)
        return max_island_size # 9


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