<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:

        def canReach(sr, sc, nsr, nsc):
            visited = set()
            q = [(sr, sc)]
            while q:
                row, col = q.pop(0)
                if (row, col) in visited: continue
                visited.add((row, col))

                if nsr == row and nsc == col:
                    return True

                for dr, dc in (0, 1), (1, 0), (0, -1), (-1, 0):
                    nr, nc = row + dr, col + dc
                    if 0 <= nr < len(grid) and 0 <= nc < len(grid[0]) and grid[nr][nc] == ".":
                        q.append((nr, nc))

            return False

        tr, tc, br, bc, sr, sc = 0, 0, 0, 0, 0, 0

        # find T, B, S
        for row in range(len(grid)):
            for col in range(len(grid[0])):
                if grid[row][col] == "T":
                    tr, tc = row, col
                    grid[row][col] = "."
                elif grid[row][col] == "B":
                    br, bc = row, col
                    grid[row][col] = "."
                elif grid[row][col] == "S":
                    sr, sc = row, col
                    grid[row][col] = "."

        # BFS start with B, and S can move to corresponding position
        visited = set()
        q = [(br, bc, sr, sc, 0)]   # row, col, step
        while q:
            row, col, sr1, sc1, step = q.pop(0)
            if (row, col, sr1, sc1) in visited: continue
            visited.add((row, col, sr1, sc1))

            if tr == row and tc == col:
                return step

            grid[row][col] = "B"
            br, bc = row, col

            for dr, dc in (0, 1), (1, 0), (0, -1), (-1, 0):
                nr, nc = row + dr, col + dc
                nsr, nsc = row - dr, col - dc
                if 0 <= nr < len(grid) and 0 <= nc < len(grid[0]) and grid[nr][nc] == "." and 0 <= nsr < len(grid) and 0 <= nsc < len(grid[0]) and grid[nsr][nsc] == "." and canReach(sr1, sc1,nsr, nsc):
                    q.append((nr, nc, row, col, step + 1))

            grid[row][col] = "."

        return -1
        # return 

'''
["#",".",".","#","T","#","#","#","#"],
["#",".",".","#",".","#",".",".","#"],
["#",".",".","#",".","#","B",".","#"],
["#",".",".",".",".",".",".",".","#"],
["#",".",".",".",".","#",".","S","#"],
["#",".",".","#",".","#","#","#","#"]]
'''
130