Find the Safest Path in a Grid
<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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2800-2899/2812.Find%20the%20Safest%20Path%20in%20a%20Grid/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2800-2899/2812.Find%20the%20Safest%20Path%20in%20a%20Grid/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2800-2899/2812.Find%20the%20Safest%20Path%20in%20a%20Grid/images/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>
Breadth-First Search; Union Find; Array; Binary Search; Matrix; Heap (Priority Queue)
use std::collections::VecDeque;
impl Solution {
    fn dfs(i: usize, j: usize, v: i32, g: &Vec<Vec<i32>>, vis: &mut Vec<Vec<bool>>) -> bool {
        if vis[i][j] || g[i][j] <= v {
            return false;
        }
        vis[i][j] = true;
        let n = g.len();
        (i == n - 1 && j == n - 1)
            || (i != 0 && Self::dfs(i - 1, j, v, g, vis))
            || (i != n - 1 && Self::dfs(i + 1, j, v, g, vis))
            || (j != 0 && Self::dfs(i, j - 1, v, g, vis))
            || (j != n - 1 && Self::dfs(i, j + 1, v, g, vis))
    }

    pub fn maximum_safeness_factor(grid: Vec<Vec<i32>>) -> i32 {
        let n = grid.len();
        let mut g = vec![vec![-1; n]; n];
        let mut q = VecDeque::new();
        for i in 0..n {
            for j in 0..n {
                if grid[i][j] == 1 {
                    q.push_back((i, j));
                }
            }
        }
        let mut level = 0;
        while !q.is_empty() {
            let m = q.len();
            for _ in 0..m {
                let (i, j) = q.pop_front().unwrap();
                if g[i][j] != -1 {
                    continue;
                }
                g[i][j] = level;
                if i != n - 1 {
                    q.push_back((i + 1, j));
                }
                if i != 0 {
                    q.push_back((i - 1, j));
                }
                if j != n - 1 {
                    q.push_back((i, j + 1));
                }
                if j != 0 {
                    q.push_back((i, j - 1));
                }
            }
            level += 1;
        }
        let mut left = 0;
        let mut right = level;
        while left < right {
            let mid = (left + right) >> 1;
            if Self::dfs(0, 0, mid, &g, &mut vec![vec![false; n]; n]) {
                left = mid + 1;
            } else {
                right = mid;
            }
        }
        right
    }
}
