<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:
        m, n = len(grid), len(grid[0])
        directions = [(0, 1), (0, -1), (1, 0), (-1, 0)]

        # Find initial positions of the player, box, and target
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 'S':
                    player = (i, j)
                elif grid[i][j] == 'B':
                    box = (i, j)
                elif grid[i][j] == 'T':
                    target = (i, j)

        # Check if a position is within bounds and not a wall
        def is_valid(x, y):
            return 0 <= x < m and 0 <= y < n and grid[x][y] != '#'

        # Check if the player can reach the position (x, y) from the current position
        def can_reach(curr_player, next_player, box):
            visited = set()
            queue = deque([curr_player])
            while queue:
                curr_x, curr_y = queue.popleft()
                if (curr_x, curr_y) == next_player:
                    return True
                for dx, dy in directions:
                    next_x, next_y = curr_x + dx, curr_y + dy
                    if is_valid(next_x, next_y) and (next_x, next_y) not in visited and (next_x, next_y) != box:
                        visited.add((next_x, next_y))
                        queue.append((next_x, next_y))
            return False

        # BFS to find the minimum number of pushes
        queue = deque([(player, box, 0)])
        visited_states = set()
        while queue:
            curr_player, curr_box, pushes = queue.popleft()
            if curr_box == target:
                return pushes
            if (curr_player, curr_box) in visited_states:
                continue
            visited_states.add((curr_player, curr_box))
            for dx, dy in directions:
                next_box_x, next_box_y = curr_box[0] + dx, curr_box[1] + dy
                next_player_x, next_player_y = curr_box[0] - dx, curr_box[1] - dy
                if is_valid(next_box_x, next_box_y) and can_reach(curr_player, (next_player_x, next_player_y), curr_box):
                    queue.append(((next_player_x, next_player_y), (next_box_x, next_box_y), pushes + 1))
        return -1

# Example usage:
solution = Solution()
grid = [["#","#","#","#","#","#"],
        ["#","T","#","#","#","#"],
        ["#",".",".","B",".","#"],
        ["#",".","#","#",".","#"],
        ["#",".",".",".","S","#"],
        ["#","#","#","#","#","#"]]
print(solution.minPushBox(grid))  # Output: 3
337