<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:
    int is_path(vector<vector<int>> &m_dist, int safeness_factor){
        int m = m_dist.size();
        int n = m_dist[0].size();
        queue<pair<int, int>> q;
        vector<vector<int>> vis(m, vector<int>(n, 0));
        if(m_dist[0][0] >= safeness_factor){
            q.push({0, 0});
            vis[0][0] = true;
        }
        
        vector<vector<int>> temp = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
        
        
        while(!q.empty()){
            int x = q.front().first;
            int y = q.front().second;
            q.pop();
            if(x == m-1 && y == n-1){
                return true;
            }
            for(int i = 0; i < 4; i++){
                int new_x = x + temp[i][0];
                int new_y = y + temp[i][1];
                // int diff = ;
                if(new_x >= 0 && new_x < m && new_y >= 0 && new_y < n && !vis[new_x][new_y] && m_dist[new_x][new_y] >= safeness_factor){
                    q.push({new_x, new_y});
                    vis[new_x][new_y] = 1;
                }
            }
        }
        return false;
    }
    int maximumSafenessFactor(vector<vector<int>>& grid) {
        int m = grid.size();
        int n = grid[0].size();
        queue<pair<int, int>> q;
        vector<vector<int>>m_dist(m, vector<int>(n, 1e9));
        for(int i = 0; i < m; i++){
            for(int j = 0; j < n; j++){
                if(grid[i][j] == 1){
                    q.push({i, j});
                    m_dist[i][j] = 0;
                }
            }
        }
        vector<vector<int>> vis(m, vector<int>(n, 0));
        vector<vector<int>> dir = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
        while(!q.empty()){
            // int size = q.size();
            // while(size--){
                int r = q.front().first;
                int c = q.front().second;
            // int dist = q.front().second;
            // m_dist[r][c] = dist;
                q.pop();
                for(int i = 0; i < 4; i++){
                    int new_r = r + dir[i][0];
                    int new_c = c + dir[i][1];
                    if(new_r >= 0 && new_c >= 0 && new_r < m && new_c < n && grid[new_r][new_c] == 0 && !vis[new_r][new_c]){
                        q.push({new_r, new_c});
                        vis[new_r][new_c] = 1;
                        m_dist[new_r][new_c] = m_dist[r][c] + 1;
                    }
                }
            // }
        }
        for(int i = 0; i < m; i++){
            for(int j = 0; j < n; j++){
                cout<<m_dist[i][j]<<" ";
            }
            cout<<endl;
        }
        // for(int i = n-1; i >= 0; i--){
        //     int safeness_factor = i;
        //     if(is_path(m_dist, safeness_factor)){
        //         return safeness_factor;
        //     }
        // }
        
        // Binary search
        int low = 0, high = n, ans;
        while(low <= high){
            int mid = (low + high)/2;
            int safeness_factor = mid;
            if(is_path(m_dist, safeness_factor)){
                ans = safeness_factor;
                low = mid + 1;
            }
            else{
                high = mid - 1;
            }
        }
        return ans;
    }
};
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