<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:
        # step 1, give each island a different tag, and calculate it's size. 
        # step 2, see how each water is connected to different island. 
        m,n = len(grid), len(grid[0])
        island_size_dict = {}

        def dfs(i,j, tag):
            if not (0<=i<m and 0<=j<n):
                return 
            
            if grid[i][j] != 1:
                return 
            
            grid[i][j] = tag 
            if tag in island_size_dict:
                island_size_dict[tag] += 1
            else:
                island_size_dict[tag] = 1
            
            directions = [[1,0],[-1,-0],[0,1],[0,-1]]
            for i_delta, j_delta in directions:
                new_i = i+i_delta 
                new_j = j+j_delta
                dfs(new_i, new_j,tag)

        tag = 10
        for i in range(m):
            for j in range(n):
                dfs(i,j, tag)
                tag += 1

        if not island_size_dict:
            return 1
            
        def find_island(i,j):
            adjacent_island = set()
            directions = [[1,0],[-1,-0],[0,1],[0,-1]]
            for i_delta, j_delta in directions:
                new_i = i+i_delta 
                new_j = j+j_delta
                if 0<=new_i<m and 0<=new_j<n and grid[new_i][new_j] != 0:
                    adjacent_island.add(grid[new_i][new_j])
            return adjacent_island

        rst = max(list(island_size_dict.values()))
        for i in range(m):
            for j in range(n):
                if grid[i][j] != 0:
                    continue 
                islands = find_island(i,j)
                tmp = 1
                for tag in islands:
                    tmp += island_size_dict[tag]
                
                rst = max(rst, tmp)

        return rst

        


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