<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)
        M = len(grid[0])
        componentId = 0
        largestSize = 0

        componentMap = dict()
        componentSize = dict()

        allOnes = True
        for i in range(N):
            for j in range(M):
                if grid[i][j] == 0:
                    allOnes = False
                    break

        if allOnes:
            return N * M

        for i in range(N):
            for j in range(M):
                if grid[i][j] == 1 and (i, j) not in componentMap:
                    visited = set()
                    size = [0]

                    def dfs(x, y):
                        if x < 0 or x >= N or y < 0 or y >= M or grid[x][y] != 1:
                            return
                        elif (x, y) in visited and (x, y) in componentMap:
                            return
                        visited.add((x, y))
                        componentMap[(x, y)] = componentId
                        size[0] += 1

                        dfs(x + 1, y)
                        dfs(x - 1, y)
                        dfs(x, y + 1)
                        dfs(x, y - 1)

                    dfs(i, j)
                    componentSize[componentId] = size[0]
                    componentId += 1

        for i in range(N):
            for j in range(M):
                if grid[i][j] == 0:
                    components = set()
                    totalSize = 1
                    if i - 1 >= 0 and grid[i - 1][j] == 1:
                        comp = componentMap[(i - 1, j)]
                        components.add(comp)
                        totalSize += componentSize[comp]
                    if i + 1 < N and grid[i + 1][j] == 1 and componentMap[(i + 1, j)] not in components:
                        comp = componentMap[(i + 1, j)]
                        components.add(comp)
                        totalSize += componentSize[comp]
                    if j + 1 < M and grid[i][j + 1] == 1 and componentMap[(i, j + 1)] not in components:
                        comp = componentMap[(i, j + 1)]
                        components.add(comp)
                        totalSize += componentSize[comp]
                    if j - 1 >= 0 and grid[i][j - 1] == 1 and componentMap[(i, j - 1)] not in components:
                        comp = componentMap[(i, j - 1)]
                        components.add(comp)
                        totalSize += componentSize[comp]

                    largestSize = max(totalSize, largestSize)

        return largestSize
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