<p>You are given two <code>m x n</code> binary matrices <code>grid1</code> and <code>grid2</code> containing only <code>0</code>&#39;s (representing water) and <code>1</code>&#39;s (representing land). An <strong>island</strong> is a group of <code>1</code>&#39;s connected <strong>4-directionally</strong> (horizontal or vertical). Any cells outside of the grid are considered water cells.</p>

<p>An island in <code>grid2</code> is considered a <strong>sub-island </strong>if there is an island in <code>grid1</code> that contains <strong>all</strong> the cells that make up <strong>this</strong> island in <code>grid2</code>.</p>

<p>Return the <em><strong>number</strong> of islands in </em><code>grid2</code> <em>that are considered <strong>sub-islands</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/10/test1.png" style="width: 493px; height: 205px;" />
<pre>
<strong>Input:</strong> grid1 = [[1,1,1,0,0],[0,1,1,1,1],[0,0,0,0,0],[1,0,0,0,0],[1,1,0,1,1]], grid2 = [[1,1,1,0,0],[0,0,1,1,1],[0,1,0,0,0],[1,0,1,1,0],[0,1,0,1,0]]
<strong>Output:</strong> 3
<strong>Explanation: </strong>In the picture above, the grid on the left is grid1 and the grid on the right is grid2.
The 1s colored red in grid2 are those considered to be part of a sub-island. There are three sub-islands.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/03/testcasex2.png" style="width: 491px; height: 201px;" />
<pre>
<strong>Input:</strong> grid1 = [[1,0,1,0,1],[1,1,1,1,1],[0,0,0,0,0],[1,1,1,1,1],[1,0,1,0,1]], grid2 = [[0,0,0,0,0],[1,1,1,1,1],[0,1,0,1,0],[0,1,0,1,0],[1,0,0,0,1]]
<strong>Output:</strong> 2 
<strong>Explanation: </strong>In the picture above, the grid on the left is grid1 and the grid on the right is grid2.
The 1s colored red in grid2 are those considered to be part of a sub-island. There are two sub-islands.
</pre>

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

<ul>
	<li><code>m == grid1.length == grid2.length</code></li>
	<li><code>n == grid1[i].length == grid2[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 500</code></li>
	<li><code>grid1[i][j]</code> and <code>grid2[i][j]</code> are either <code>0</code> or <code>1</code>.</li>
</ul>

class Solution:
    # Directions in which we can traverse inside the grids.
    directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]

    # Helper method to check if the cell at the position (x, y) in the 'grid'
    # is a land cell.
    def is_cell_land(self, x, y, grid):
        return grid[x][y] == 1

    # Traverse all cells of island starting at position (x, y) in 'grid2',
    # and check if this island is a sub-island in 'grid1'.
    def is_sub_island(self, x, y, grid1, grid2, visited):
        total_rows = len(grid2)
        total_cols = len(grid2[0])
        # Traverse on all cells using the depth-first search method.
        is_sub_island = True

        # If the current cell is not a land cell in 'grid1', then the current island can't be a sub-island.
        if not self.is_cell_land(x, y, grid1):
            is_sub_island = False

        # Traverse on all adjacent cells.
        for direction in self.directions:
            next_x = x + direction[0]
            next_y = y + direction[1]
            # If the next cell is inside 'grid2', is not visited, and is a land cell,
            # then we traverse to the next cell.
            if (
                0 <= next_x < total_rows
                and 0 <= next_y < total_cols
                and not visited[next_x][next_y]
                and self.is_cell_land(next_x, next_y, grid2)
            ):
                # Mark the next cell as visited.
                visited[next_x][next_y] = True
                next_cell_is_part_of_sub_island = self.is_sub_island(
                    next_x, next_y, grid1, grid2, visited
                )
                is_sub_island = (
                    is_sub_island and next_cell_is_part_of_sub_island
                )
        return is_sub_island

    def countSubIslands(self, grid1: List[List[int]], grid2: List[List[int]]) -> int:
        total_rows = len(grid2)
        total_cols = len(grid2[0])

        visited = [[False] * total_cols for _ in range(total_rows)]
        sub_island_counts = 0

        # Iterate over each cell in 'grid2'.
        for x in range(total_rows):
            for y in range(total_cols):
                # If the cell at position (x, y) in 'grid2' is not visited,
                # is a land cell in 'grid2', and the island starting from this cell is a sub-island in 'grid1',
                # then increment the count of sub-islands.
                if not visited[x][y] and self.is_cell_land(x, y, grid2):
                    visited[x][y] = True
                    if self.is_sub_island(x, y, grid1, grid2, visited):
                        sub_island_counts += 1

        # Return total count of sub-islands.
        return sub_island_counts
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