<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:

    # Directions for moving to neighboring cells: right, left, down, up

    dir = [(0, 1), (0, -1), (1, 0), (-1, 0)]

    def maximumSafenessFactor(self, grid: List[List[int]]) -> int:

        n = len(grid)

        multi_source_queue = deque()

        # Mark thieves as 0 and empty cells as -1, and push thieves to the queue

        for i in range(n):

            for j in range(n):

                if grid[i][j] == 1:

                    # Push thief coordinates to the queue

                    multi_source_queue.append((i, j))

                    # Mark thief cell with 0

                    grid[i][j] = 0

                else:

                    # Mark empty cell with -1

                    grid[i][j] = -1

        # Calculate safeness factor for each cell using BFS

        while multi_source_queue:

            size = len(multi_source_queue)

            while size > 0:

                curr = multi_source_queue.popleft()

                # Check neighboring cells

                for d in self.dir:

                    di, dj = curr[0] + d[0], curr[1] + d[1]

                    val = grid[curr[0]][curr[1]]

                    # Check if the cell is valid and unvisited

                    if self.isValidCell(grid, di, dj) and grid[di][dj] == -1:

                        # Update safeness factor and push to the queue

                        grid[di][dj] = val + 1

                        multi_source_queue.append((di, dj))

                size -= 1

        # Binary search for maximum safeness factor

        start, end, res = 0, 0, -1

        for i in range(n):

            for j in range(n):

                # Set end as the maximum safeness factor possible

                end = max(end, grid[i][j])

        while start <= end:

            mid = start + (end - start) // 2

            if self.isValidSafeness(grid, mid):

                # Store valid safeness and search for larger ones

                res = mid

                start = mid + 1

            else:

                end = mid - 1

        return res

    

    # Check if a given cell lies within the grid

    def isValidCell(self, grid, i, j) -> bool:

        n = len(grid)

        return 0 <= i < n and 0 <= j < n

    # Check if a path exists with given minimum safeness value

    def isValidSafeness(self, grid, min_safeness) -> bool:

        n = len(grid)

        # Check if the source and destination cells satisfy minimum safeness

        if grid[0][0] < min_safeness or grid[n - 1][n - 1] < min_safeness:

            return False

        traversal_queue = deque([(0, 0)])

        visited = [[False] * n for _ in range(n)]

        visited[0][0] = True

        # Breadth-first search to find a valid path

        while traversal_queue:

            curr = traversal_queue.popleft()

            if curr[0] == n - 1 and curr[1] == n - 1:

                return True  # Valid path found

            # Check neighboring cells

            for d in self.dir:

                di, dj = curr[0] + d[0], curr[1] + d[1]

                # Check if the neighboring cell is valid and unvisited

                if self.isValidCell(grid, di, dj) and not visited[di][dj] and grid[di][dj] >= min_safeness:

                    visited[di][dj] = True

                    traversal_queue.append((di, dj))

        return False  # No valid path found
    
        
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