<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>

class Solution:
    def minPushBox(self, grid: List[List[str]]) -> int:
        m,n=len(grid), len(grid[0])
        for i in range(m):
            for j in range(n):
                if grid[i][j]=='T':
                    target=(i,j)
                if grid[i][j]=='B':
                    box = (i,j)
                if grid[i][j]=='S':
                    person = (i,j)
        
        def valid(x,y):
            return 0<=x<m and 0<=y<n and grid[x][y]!='#'
        
        def can_person_travel(curr,dest,box):
            q = deque([curr])
            v = set()
            while q:
                pos = q.popleft()
                if pos==dest:
                    return True
                
                new_positions = [(pos[0]+1,pos[1]),(pos[0]-1,pos[1]),(pos[0],pos[1]+1),(pos[0],pos[1]-1)]
                for x,y in new_positions:
                    if valid(x,y) and (x,y) not in v and (x,y)!=box:
                        v.add((x,y))
                        q.append((x,y))
            
            return False
        
        dq = deque([(0,box,person)])
        visited = {box+person}
        while dq:
            dist, box, person = dq.popleft()
            if box==target:
                return dist
            
            b_moves = [(box[0]+1, box[1]),
                       (box[0]-1, box[1]),
                       (box[0], box[1]+1),
                       (box[0], box[1]-1)]
            
            p_moves = [(box[0]-1, box[1]),
                       (box[0]+1, box[1]),
                       (box[0], box[1]-1),
                       (box[0], box[1]+1)]
            
            for new_box, new_person in zip(b_moves, p_moves):
                if valid(new_box[0],new_box[1]) and new_box+box not in visited:
                    if valid(new_person[0],new_person[1]) and can_person_travel(person, new_person, box):
                        visited.add(new_box+box)
                        dq.append((dist+1, new_box, box))
        
        return -1
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