<p>A storekeeper is a game in which the player pushes boxes around in a warehouse trying to get them to target locations.</p>

<p>The game is represented by an <code>m x n</code> grid of characters <code>grid</code> where each element is a wall, floor, or box.</p>

<p>Your task is to move the box <code>&#39;B&#39;</code> to the target position <code>&#39;T&#39;</code> under the following rules:</p>

<ul>
	<li>The character <code>&#39;S&#39;</code> represents the player. The player can move up, down, left, right in <code>grid</code> if it is a floor (empty cell).</li>
	<li>The character <code>&#39;.&#39;</code> represents the floor which means a free cell to walk.</li>
	<li>The character<font face="monospace">&nbsp;</font><code>&#39;#&#39;</code><font face="monospace">&nbsp;</font>represents the wall which means an obstacle (impossible to walk there).</li>
	<li>There is only one box <code>&#39;B&#39;</code> and one target cell <code>&#39;T&#39;</code> in the <code>grid</code>.</li>
	<li>The box can be moved to an adjacent free cell by standing next to the box and then moving in the direction of the box. This is a <strong>push</strong>.</li>
	<li>The player cannot walk through the box.</li>
</ul>

<p>Return <em>the minimum number of <strong>pushes</strong> to move the box to the target</em>. If there is no way to reach the target, return <code>-1</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2019/11/06/sample_1_1620.png" style="width: 500px; height: 335px;" />
<pre>
<strong>Input:</strong> grid = [[&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;T&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;B&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;#&quot;,&quot;#&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;S&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> We return only the number of times the box is pushed.</pre>

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

<pre>
<strong>Input:</strong> grid = [[&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;T&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;B&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;S&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;]]
<strong>Output:</strong> -1
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> grid = [[&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;T&quot;,&quot;.&quot;,&quot;.&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;#&quot;,&quot;B&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;S&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;]]
<strong>Output:</strong> 5
<strong>Explanation:</strong> push the box down, left, left, up and up.
</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;= 20</code></li>
	<li><code>grid</code> contains only characters <code>&#39;.&#39;</code>, <code>&#39;#&#39;</code>, <code>&#39;S&#39;</code>, <code>&#39;T&#39;</code>, or <code>&#39;B&#39;</code>.</li>
	<li>There is only one character <code>&#39;S&#39;</code>, <code>&#39;B&#39;</code>, and <code>&#39;T&#39;</code> in the <code>grid</code>.</li>
</ul>

from collections import deque


class Solution:

    def minPushBox(self, grid: List[List[str]]) -> int:

      

        # Helper function to flatten the 2D grid coordinates into 1D.

        def get_flat_index(row: int, col: int) -> int:

            return row * cols + col


        # Check if the position (i, j) is within the grid and not an obstacle.

        def is_valid(row: int, col: int) -> bool:

            return 0 <= row < rows and 0 <= col < cols and grid[row][col] != "#"


        # Find the starting positions of the Player (S) and the Box (B).

        for i, row in enumerate(grid):

            for j, cell in enumerate(row):

                if cell == "S":

                    start_row, start_col = i, j

                elif cell == "B":

                    box_row, box_col = i, j


        # Dimensions of the grid

        rows, cols = len(grid), len(grid[0])


        # Possible 4 directions (up, right, down, left) as delta coordinates.

        directions = (-1, 0, 1, 0, -1)


        # Initialize the queue with the starting position of the player and box, with a move distance of 0.

        queue = deque([(get_flat_index(start_row, start_col), get_flat_index(box_row, box_col), 0)])


        # Visited positions: Visited states are represented by player and box positions.

        visited = [[False] * (rows * cols) for _ in range(rows * cols)]

        visited[get_flat_index(start_row, start_col)][get_flat_index(box_row, box_col)] = True


        while queue:

            player, box, distance = queue.popleft()

            box_row, box_col = box // cols, box % cols


            # If the current box position is the target, return the distance.

            if grid[box_row][box_col] == "T":

                return distance


            player_row, player_col = player // cols, player % cols


            # Iterate over all possible moves.

            for delta_row, delta_col in zip(directions, directions[1:]):

                next_player_row, next_player_col = player_row + delta_row, player_col + delta_col


                # Check if the next player position is valid.

                if not is_valid(next_player_row, next_player_col):

                    continue

              

                # The player moves the box if they are adjacent.

                if next_player_row == box_row and next_player_col == box_col:

                    next_box_row, next_box_col = box_row + delta_row, box_col + delta_col

                  

                    # Check if the new box position is valid and has not been visited.

                    if not is_valid(next_box_row, next_box_col) or visited[get_flat_index(next_player_row, next_player_col)][get_flat_index(next_box_row, next_box_col)]:

                        continue


                    # Update visited and add the new state to the queue.

                    visited[get_flat_index(next_player_row, next_player_col)][get_flat_index(next_box_row, next_box_col)] = True

                    queue.append((get_flat_index(next_player_row, next_player_col), get_flat_index(next_box_row, next_box_col), distance + 1))

              

                # If the player is not pushing the box, we add the new state to the front of the queue to check it sooner.

                elif not visited[get_flat_index(next_player_row, next_player_col)][get_flat_index(box_row, box_col)]:

                    visited[get_flat_index(next_player_row, next_player_col)][get_flat_index(box_row, box_col)] = True

                    queue.appendleft((get_flat_index(next_player_row, next_player_col), get_flat_index(box_row, box_col), distance))


        # Return -1 if it's impossible to push the box to the target.

        return -1
289