<p>You are given a <strong>positive</strong> integer <code>k</code>. You are also given:</p>

<ul>
	<li>a 2D integer array <code>rowConditions</code> of size <code>n</code> where <code>rowConditions[i] = [above<sub>i</sub>, below<sub>i</sub>]</code>, and</li>
	<li>a 2D integer array <code>colConditions</code> of size <code>m</code> where <code>colConditions[i] = [left<sub>i</sub>, right<sub>i</sub>]</code>.</li>
</ul>

<p>The two arrays contain integers from <code>1</code> to <code>k</code>.</p>

<p>You have to build a <code>k x k</code> matrix that contains each of the numbers from <code>1</code> to <code>k</code> <strong>exactly once</strong>. The remaining cells should have the value <code>0</code>.</p>

<p>The matrix should also satisfy the following conditions:</p>

<ul>
	<li>The number <code>above<sub>i</sub></code> should appear in a <strong>row</strong> that is strictly <strong>above</strong> the row at which the number <code>below<sub>i</sub></code> appears for all <code>i</code> from <code>0</code> to <code>n - 1</code>.</li>
	<li>The number <code>left<sub>i</sub></code> should appear in a <strong>column</strong> that is strictly <strong>left</strong> of the column at which the number <code>right<sub>i</sub></code> appears for all <code>i</code> from <code>0</code> to <code>m - 1</code>.</li>
</ul>

<p>Return <em><strong>any</strong> matrix that satisfies the conditions</em>. If no answer exists, return an empty matrix.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/07/06/gridosdrawio.png" style="width: 211px; height: 211px;" />
<pre>
<strong>Input:</strong> k = 3, rowConditions = [[1,2],[3,2]], colConditions = [[2,1],[3,2]]
<strong>Output:</strong> [[3,0,0],[0,0,1],[0,2,0]]
<strong>Explanation:</strong> The diagram above shows a valid example of a matrix that satisfies all the conditions.
The row conditions are the following:
- Number 1 is in row <u>1</u>, and number 2 is in row <u>2</u>, so 1 is above 2 in the matrix.
- Number 3 is in row <u>0</u>, and number 2 is in row <u>2</u>, so 3 is above 2 in the matrix.
The column conditions are the following:
- Number 2 is in column <u>1</u>, and number 1 is in column <u>2</u>, so 2 is left of 1 in the matrix.
- Number 3 is in column <u>0</u>, and number 2 is in column <u>1</u>, so 3 is left of 2 in the matrix.
Note that there may be multiple correct answers.
</pre>

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

<pre>
<strong>Input:</strong> k = 3, rowConditions = [[1,2],[2,3],[3,1],[2,3]], colConditions = [[2,1]]
<strong>Output:</strong> []
<strong>Explanation:</strong> From the first two conditions, 3 has to be below 1 but the third conditions needs 3 to be above 1 to be satisfied.
No matrix can satisfy all the conditions, so we return the empty matrix.
</pre>

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

<ul>
	<li><code>2 &lt;= k &lt;= 400</code></li>
	<li><code>1 &lt;= rowConditions.length, colConditions.length &lt;= 10<sup>4</sup></code></li>
	<li><code>rowConditions[i].length == colConditions[i].length == 2</code></li>
	<li><code>1 &lt;= above<sub>i</sub>, below<sub>i</sub>, left<sub>i</sub>, right<sub>i</sub> &lt;= k</code></li>
	<li><code>above<sub>i</sub> != below<sub>i</sub></code></li>
	<li><code>left<sub>i</sub> != right<sub>i</sub></code></li>
</ul>

class Solution {
private:
    class LoopTest{
    private:
        const int k;
        vector<vector<bool>> m;
        bool valid_;
        vector<int> poses;  
        vector<int> prior;  
        int maxprio;           
    public:
        LoopTest(int K):k(K),prior(K+1,-1),poses(K+1,1),maxprio(-1){
            m.reserve(k);
            for(int i = 0;i<k;i++)
                m.emplace_back(k,false);            
            valid_ = true;
        };
        int GetPos(int n){
            return poses[n];
        }
        int GetPrio(int n,vector<bool> &visited){            
            if(prior[n]>=0) return prior[n];
            valid_ = valid_ && !visited[n];
            if(!valid_) return 0;
            visited[n]=true;            
            n--;            
            int pr = 0;
            for(int i = 0;i<k;i++){
                if(m[n][i]){                    
                    pr = max(pr,GetPrio(i+1,visited)+1);                    
                }
            }
            prior[n+1] = pr;
            visited[n+1]=false;            
            return pr;
        }      
        void CalcPoses(){
            vector<bool> visited(k+1,false);
            for(int i = 1;i<=k;i++){                
                GetPrio(i,visited);                                                
            }                        
            int pos = k;
            vector<int> sorted;            
            sorted.reserve(k+1);
            for(int i = 0;i<=k;i++)
                sorted.push_back(i);
            std::sort(sorted.begin()+1,sorted.end(),[&](int a, int b){return prior[a]<prior[b];});

            for(int i = 1;i<=k;i++){
                poses[sorted[i]] = pos;
                pos--;
            }            
        }
        bool push(int f, int t){
            f--;t--;
            m[f][t] = true;            
            return true;
        }
        bool Valide(){
            return valid_;
        }

    };
    struct RC{
        int r = 0;
        int c = 0;
    };
public:
    vector<vector<int>> buildMatrix(int k, vector<vector<int>>& rowConditions, vector<vector<int>>& colConditions) {
        LoopTest RL(k);                
        for(auto &a:rowConditions)
            RL.push(a[0],a[1]);                

        LoopTest CL(k);
        for(auto &a:colConditions)
            CL.push(a[0],a[1]);

        RL.CalcPoses();
        CL.CalcPoses();
        if(!RL.Valide() || !CL.Valide())
            return {};
        vector<vector<int>> vec;
        vec.reserve(k);
        for(int i = 0;i<k;i++){
            vec.push_back(vector<int>(k,0));
        }
        for(int i = 1;i<=k;i++){
            vec[RL.GetPos(i)-1][CL.GetPos(i)-1] = i;
            //vec[i-1][i-1] = i;
        }
        return vec;
    }
};
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