<p>Given an <code>m x n</code> 2D binary grid <code>grid</code> which represents a map of <code>&#39;1&#39;</code>s (land) and <code>&#39;0&#39;</code>s (water), return <em>the number of islands</em>.</p>

<p>An <strong>island</strong> is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are all surrounded by water.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> grid = [
  [&quot;1&quot;,&quot;1&quot;,&quot;1&quot;,&quot;1&quot;,&quot;0&quot;],
  [&quot;1&quot;,&quot;1&quot;,&quot;0&quot;,&quot;1&quot;,&quot;0&quot;],
  [&quot;1&quot;,&quot;1&quot;,&quot;0&quot;,&quot;0&quot;,&quot;0&quot;],
  [&quot;0&quot;,&quot;0&quot;,&quot;0&quot;,&quot;0&quot;,&quot;0&quot;]
]
<strong>Output:</strong> 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [
  [&quot;1&quot;,&quot;1&quot;,&quot;0&quot;,&quot;0&quot;,&quot;0&quot;],
  [&quot;1&quot;,&quot;1&quot;,&quot;0&quot;,&quot;0&quot;,&quot;0&quot;],
  [&quot;0&quot;,&quot;0&quot;,&quot;1&quot;,&quot;0&quot;,&quot;0&quot;],
  [&quot;0&quot;,&quot;0&quot;,&quot;0&quot;,&quot;1&quot;,&quot;1&quot;]
]
<strong>Output:</strong> 3
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 300</code></li>
	<li><code>grid[i][j]</code> is <code>&#39;0&#39;</code> or <code>&#39;1&#39;</code>.</li>
</ul>

class Solution:
    def numIslands(self, grid: List[List[str]]) -> int:
        m, n = len(grid), len(grid[0])

        rank_matrix = [[1 if grid[i][j] == '1' else 0 for j in range(n)] for i in range(m)]
        parent_matrix = [[(i, j) if grid[i][j] == '1' else (-1, -1) for j in range(n)] for i in range(m)] 
        #print(parent_matrix)

        def find(i, j):
            cur_i, cur_j = i, j
            while parent_matrix[cur_i][cur_j] != (cur_i, cur_j):
                # print(parent_matrix[cur_i][cur_j])
                cur_i, cur_j = parent_matrix[cur_i][cur_j]
            return cur_i, cur_j
        
        def union(i1, j1, i2, j2):
            p1i, p1j = find(i1, j1)
            p2i, p2j = find(i2, j2)
            if p1i != p2i or p1j != p2j:
                if rank_matrix[p1i][p1j] < rank_matrix[p2i][p2j]:
                    p1i, p1j, p2i, p2j = p2i, p2j, p1i, p1j
                parent_matrix[p2i][p2j] = parent_matrix[p1i][p1j]
                rank_matrix[p1i][p1j] += rank_matrix[p2i][p2j]
                rank_matrix[p2i][p2j] = 0
        
        for i in range(m):
            for j in range(n):
                for y, x in [(0, 1), (1, 0), (-1, 0), (0, -1)]:
                    if 0 <= (i + y) < m and 0 <= (j + x) < n and grid[i + y][j + x] == '1' and grid[i][j] == '1':
                        union(i, j, i + y, j + x)
        cnt = 0
        for i in range(m):
            for j in range(n):
                if parent_matrix[i][j] == (i, j):
                    cnt += 1
        # print(parent_matrix)
        return cnt
                
            
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