<p>The <strong>n-queens</strong> puzzle is the problem of placing <code>n</code> queens on an <code>n x n</code> chessboard such that no two queens attack each other.</p>

<p>Given an integer <code>n</code>, return <em>all distinct solutions to the <strong>n-queens puzzle</strong></em>. You may return the answer in <strong>any order</strong>.</p>

<p>Each solution contains a distinct board configuration of the n-queens&#39; placement, where <code>&#39;Q&#39;</code> and <code>&#39;.&#39;</code> both indicate a queen and an empty space, respectively.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/13/queens.jpg" style="width: 600px; height: 268px;" />
<pre>
<strong>Input:</strong> n = 4
<strong>Output:</strong> [[&quot;.Q..&quot;,&quot;...Q&quot;,&quot;Q...&quot;,&quot;..Q.&quot;],[&quot;..Q.&quot;,&quot;Q...&quot;,&quot;...Q&quot;,&quot;.Q..&quot;]]
<strong>Explanation:</strong> There exist two distinct solutions to the 4-queens puzzle as shown above
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> [[&quot;Q&quot;]]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 9</code></li>
</ul>

class Solution {
public:
    vector<vector<string>> solveNQueens(int n) {
        vector<vector<string>> ans;
        string dots(n, '.');
        vector<string> graph(n, dots);
        vector<vector<bool>> trace(n, vector<bool>(n, true));
        backTrack(ans, graph, trace, n, 0);
        return ans;
    }
private:
    void backTrack(vector<vector<string>>& ans, vector<string> graph, vector<vector<bool>> trace, int n, int idx) {
        if(idx == n) {
            ans.push_back(graph);
            return;
        }
        
        for(int j=0; j<n; j++) {
            if(trace[idx][j] == true) {
                vector<vector<bool>> tempTrace = trace;
                // vector<string> tempGraph = graph;
                updateTrace(tempTrace, idx, j);
                graph[idx][j] = 'Q';
                backTrack(ans, graph, tempTrace, n, idx+1);
                graph[idx][j] = '.';
            }
        }
    }
    void updateTrace(vector<vector<bool>>& trace, int i, int j) {
        int n=trace[0].size();
        trace[i][j] = false;
        for(int k=0; k<n; k++) {
            trace[i][k] = trace[k][j] = false;
        }
        int x=i-1, y=j-1;
        while(x > -1 && y > -1) {
            trace[x][y] = false;
            x--; y--;
        }
        x=i-1; y=j+1;
        while(x > -1 && y < n) {
            trace[x][y] = false;
            x--; y++;
        }
        x=i+1; y=j-1;
        while(x < n && y > -1) {
            trace[x][y] = false;
            x++; y--;
        }
        x=i+1; y=j+1;
        while(x < n && y < n) {
            trace[x][y] = false;
            x++; y++;
        }
        return;
    }
};
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