<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 {
public:
    bool is_neighbor(int x, int y, int n) {
        return x >= 0 && x < n && y >= 0 && y < n;
    }
    int bfs(vector<vector<int>> grid) {

        int n = grid.size();
        priority_queue<pair<int, pair<int, int>>> Q;
        pair<int, int> q;
        pair<int, int> e{n - 1, n - 1};
        int min_d, x, y;
        vector<vector<bool>> visited(n, vector<bool>(n));
        vector<pair<int, int>> dir{{0, 1}, {0, -1}, {1, 0}, {-1, 0}};

        Q.push({grid[0][0], {0, 0}});
        visited[0][0] = true;
        while (!Q.empty()) {
            min_d = Q.top().first;
            q = Q.top().second;
            cout << min_d << ' ' << q.first << ' ' << q.second << endl;

            Q.pop();

            if (q == e) {
                return min_d - 1;
            }
            for (auto d : dir) {
                x = q.first + d.first;
                y = q.second + d.second;
                if (is_neighbor(x, y, n)) {
                    if (!visited[x][y]) {
                        visited[x][y] = true;
                        Q.push({min(min_d, grid[x][y]), {x, y}});
                    }
                }
            }
        }
        return 0;
    }
    void label_shortest_dist(vector<vector<int>>& grid) {
        int x, y;
        int n = grid.size();

        queue<pair<int, int>> Q;
        pair<int, int> q;

        vector<pair<int, int>> dir{{0, 1}, {0, -1}, {1, 0}, {-1, 0}};

        for (int i = 0; i < n; i++)
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 1) {
                    Q.push({i, j});
                }
            }

        while (!Q.empty()) {
            q = Q.front();
            Q.pop();

            for (auto d : dir) {
                x = q.first + d.first;
                y = q.second + d.second;
                if (is_neighbor(x, y, n)) {
                    if (grid[x][y] == 0) {
                        grid[x][y] = grid[q.first][q.second] + 1;
                        Q.push({x, y});
                    }
                }
            }
        }
    }
    int maximumSafenessFactor(vector<vector<int>>& grid) {

        int n = grid.size();
        if (grid[0][0] == 1 || grid[n - 1][n - 1] == 1) {
            return 0;
        }

        label_shortest_dist(grid);

        return bfs(grid);
    }
};
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