<p>You are given a <strong>0-indexed</strong> 2D matrix <code>grid</code> of size <code>n x n</code>, where <code>(r, c)</code> represents:</p>

<ul>
	<li>A cell containing a thief if <code>grid[r][c] = 1</code></li>
	<li>An empty cell if <code>grid[r][c] = 0</code></li>
</ul>

<p>You are initially positioned at cell <code>(0, 0)</code>. In one move, you can move to any adjacent cell in the grid, including cells containing thieves.</p>

<p>The <strong>safeness factor</strong> of a path on the grid is defined as the <strong>minimum</strong> manhattan distance from any cell in the path to any thief in the grid.</p>

<p>Return <em>the <strong>maximum safeness factor</strong> of all paths leading to cell </em><code>(n - 1, n - 1)</code><em>.</em></p>

<p>An <strong>adjacent</strong> cell of cell <code>(r, c)</code>, is one of the cells <code>(r, c + 1)</code>, <code>(r, c - 1)</code>, <code>(r + 1, c)</code> and <code>(r - 1, c)</code> if it exists.</p>

<p>The <strong>Manhattan distance</strong> between two cells <code>(a, b)</code> and <code>(x, y)</code> is equal to <code>|a - x| + |b - y|</code>, where <code>|val|</code> denotes the absolute value of val.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/02/example1.png" style="width: 362px; height: 242px;" />
<pre>
<strong>Input:</strong> grid = [[1,0,0],[0,0,0],[0,0,1]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> All paths from (0, 0) to (n - 1, n - 1) go through the thieves in cells (0, 0) and (n - 1, n - 1).
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/02/example2.png" style="width: 362px; height: 242px;" />
<pre>
<strong>Input:</strong> grid = [[0,0,1],[0,0,0],[0,0,0]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The path depicted in the picture above has a safeness factor of 2 since:
- The closest cell of the path to the thief at cell (0, 2) is cell (0, 0). The distance between them is | 0 - 0 | + | 0 - 2 | = 2.
It can be shown that there are no other paths with a higher safeness factor.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/02/example3.png" style="width: 362px; height: 242px;" />
<pre>
<strong>Input:</strong> grid = [[0,0,0,1],[0,0,0,0],[0,0,0,0],[1,0,0,0]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The path depicted in the picture above has a safeness factor of 2 since:
- The closest cell of the path to the thief at cell (0, 3) is cell (1, 2). The distance between them is | 0 - 1 | + | 3 - 2 | = 2.
- The closest cell of the path to the thief at cell (3, 0) is cell (3, 2). The distance between them is | 3 - 3 | + | 0 - 2 | = 2.
It can be shown that there are no other paths with a higher safeness factor.
</pre>

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

<ul>
	<li><code>1 &lt;= grid.length == n &lt;= 400</code></li>
	<li><code>grid[i].length == n</code></li>
	<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
	<li>There is at least one thief in the <code>grid</code>.</li>
</ul>

class Solution:
  def maximumSafenessFactor(self, grid: List[List[int]]) -> int:
    self.dirs = ((0, 1), (1, 0), (0, -1), (-1, 0))
    n = len(grid)
    distToThief = self._getDistToThief(grid)

    def hasValidPath(safeness: int) -> bool:
      if distToThief[0][0] < safeness:
        return False

      q = collections.deque([(0, 0)])
      seen = {(0, 0)}

      while q:
        i, j = q.popleft()
        if distToThief[i][j] < safeness:
          continue
        if i == n - 1 and j == n - 1:
          return True
        for dx, dy in self.dirs:
          x = i + dx
          y = j + dy
          if x < 0 or x == n or y < 0 or y == n:
            continue
          if (x, y) in seen:
            continue
          q.append((x, y))
          seen.add((x, y))

      return False

    return bisect.bisect_left(range(n * 2), True,
                              key=lambda m: not hasValidPath(m)) - 1

  def _getDistToThief(self, grid: List[List[int]]) -> List[List[int]]:
    n = len(grid)
    distToThief = [[0] * n for _ in range(n)]
    q = collections.deque()
    seen = set()

    for i in range(n):
      for j in range(n):
        if grid[i][j] == 1:
          q.append((i, j))
          seen.add((i, j))

    dist = 0
    while q:
      for _ in range(len(q)):
        i, j = q.popleft()
        distToThief[i][j] = dist
        for dx, dy in self.dirs:
          x = i + dx
          y = j + dy
          if x < 0 or x == n or y < 0 or y == n:
            continue
          if (x, y) in seen:
            continue
          q.append((x, y))
          seen.add((x, y))
      dist += 1

    return distToThief
6788