<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:
        num_rows = len(grid)
        num_cols = len(grid[0])
        thief_locations = []
        for row_index in range(num_rows):
            for col_index in range(num_cols):
                if grid[row_index][col_index] == 1:
                    thief_locations.append((row_index, col_index, 0))
        distance_from_cell_to_thief = [
            [float("inf") for _ in range(num_cols)] for _ in range(num_rows)
        ]
        queue = deque(thief_locations)
        visited = set()
        while queue:
            current_row, current_col, current_distance_from_thief = queue.popleft()
            if (current_row, current_col) not in visited:
                visited.add((current_row, current_col))
                distance_from_cell_to_thief[current_row][
                    current_col
                ] = current_distance_from_thief
                if current_row + 1 < num_rows:
                    queue.append(
                        (current_row + 1, current_col, current_distance_from_thief + 1)
                    )
                if current_col + 1 < num_cols:
                    queue.append(
                        (current_row, current_col + 1, current_distance_from_thief + 1)
                    )
                if current_row - 1 >= 0:
                    queue.append(
                        (current_row - 1, current_col, current_distance_from_thief + 1)
                    )
                if current_col - 1 >= 0:
                    queue.append(
                        (current_row, current_col - 1, current_distance_from_thief + 1)
                    )
        low = 0
        high = float("-inf")
        for row_index in range(num_rows):
            for col_index in range(num_cols):
                high = max(high, distance_from_cell_to_thief[row_index][col_index])
        while low <= high:
            mid = (low + high) // 2
            queue = deque([(0, 0)])
            visited = set()
            is_connected = False
            while queue:
                current_row, current_column = queue.popleft()
                if (current_row, current_column) not in visited:
                    visited.add((current_row, current_column))
                    if distance_from_cell_to_thief[current_row][current_column] < mid:
                        continue
                    if current_row == num_rows - 1 and current_column == num_cols - 1:
                        is_connected = True
                        break
                    if (
                        current_row + 1 < num_rows
                        and (current_row + 1, current_column) not in visited
                    ):
                        queue.append(
                            (
                                current_row + 1,
                                current_column,
                            )
                        )
                    if (
                        current_column + 1 < num_cols
                        and (current_row, current_column + 1) not in visited
                    ):
                        queue.append(
                            (
                                current_row,
                                current_column + 1,
                            )
                        )
                    if (
                        current_row - 1 >= 0
                        and (current_row - 1, current_column) not in visited
                    ):
                        queue.append(
                            (
                                current_row - 1,
                                current_column,
                            )
                        )
                    if (
                        current_column - 1 >= 0
                        and (current_row, current_column - 1) not in visited
                    ):
                        queue.append(
                            (
                                current_row,
                                current_column - 1,
                            )
                        )
            if is_connected:
                low = mid + 1
            else:
                high = mid - 1
        return high


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