<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:
        n = len(grid)
        dp = [[(0, 0) for _ in range(n)] for _ in range(n)]
        """
        dp = [[0 for _ in range(n)] for _ in range(n)]
        
        dp = []
        for i in range(n):
            tmp = []
            for j in range(n):
                tmp.append(grid[i][j])
            dp.append(tmp)
        """
        max_large_island = 0
        visit = set()
        island_num = 0
        def find_island_cells(i, j, island_cells):
            if i >= n or j >= n or i < 0 or j < 0 or grid[i][j] != 1 or (i, j) in visit:
                return 0
            island_cells.append([i, j])
            visit.add((i, j))
            c = 1
            c += find_island_cells(i+1, j, island_cells)
            c += find_island_cells(i-1, j, island_cells)
            c += find_island_cells(i, j+1, island_cells)
            c += find_island_cells(i, j-1, island_cells)
            return c
            
        def fill_dp(island_cells: list[list[int]], count: int, island_num: int):
            nonlocal dp
            for x, y in island_cells:
                dp[x][y] = (island_num, count)
            return
        
        def check_large_island_possibility(i: int, j: int):
            nonlocal max_large_island
            g = {}
            directions = [[0, 1], [0, -1], [1, 0], [-1, 0]]
            for x, y in directions:
                newi, newj = x+i, y+j
                if 0 <= newi < n and 0 <= newj < n and grid[newi][newj] == 1:
                    (island_num, count) = dp[newi][newj]
                    if island_num not in g:
                        g[island_num] = count
                    else:
                        g[island_num] = max(g[island_num], count)
            possibility = 1
            # print(i, j)
            # print("x", g)
            for key in g:
                possibility += g[key]
            max_large_island = max(max_large_island, possibility)
            return
        
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 1 and (i, j) not in visit:
                    island_cells = []
                    island_num += 1
                    count = find_island_cells(i, j, island_cells)
                    fill_dp(island_cells, len(island_cells), island_num)
                    max_large_island = max(max_large_island, count)
        
        # print(dp)
        for i in range(n):
            for j in range(n):
                if grid[i][j] == 0:
                    check_large_island_possibility(i, j)
        
        return max_large_island
                    
        
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