<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 {
    struct TState
    {
        TState(int16_t k)
            : Children(k + 1)
            , Deg(k + 1)
        { }

        void Reset()
        {
            for (auto& c : Children) {
                c.clear();
            }

            fill(Deg.begin(), Deg.end(), 0);
        }

        vector<vector<int16_t>> Children;
        vector<int16_t> Deg;
        vector<int16_t> Layer;
        vector<int16_t> NextLayer;
    };

    static vector<int16_t> TopoSort(const vector<vector<int>>& edges, int16_t k, TState& st)
    {
        vector<int16_t> order;
        for (const auto& v : edges) {
            st.Children[v[0]].push_back(v[1]);
            ++st.Deg[v[1]];
        }

        for (int16_t i = 1; i <= k; ++i) {
            if (st.Deg[i] == 0) {
                st.Layer.push_back(i);
            }
        }

        order.reserve(k);
        while(!st.Layer.empty()) {
            for (auto v : st.Layer) {
                order.push_back(v);
                --k;
                for (auto c: st.Children[v]) {
                    if (--st.Deg[c] == 0) {
                        st.NextLayer.push_back(c);
                    }
                }
            }

            st.NextLayer.swap(st.Layer);
            st.NextLayer.clear();
        }

        if (k != 0) {
            return {};
        }

        return order;
    }
public:
    vector<vector<int>> buildMatrix(int k, vector<vector<int>>& rowConditions, vector<vector<int>>& colConditions) {
        vector<int16_t> ro, invCo;
        {
            TState st(k);
            ro = TopoSort(rowConditions, k, st);
            if (ro.empty()) {
                return {};
            }

            st.Reset();
            const auto co = TopoSort(colConditions, k, st);
            if (co.empty()) {
                return {};
            }

            invCo = std::move(st.Deg);
            invCo.assign(k + 1, 0);
            for (int16_t i = 0; i < k; ++i) {
                invCo[co[i]] = i;
            }
        }

        vector<vector<int>> res(k);
        for (auto& r : res) {
            r.resize(k);
        }

        for (int16_t i = 0; i < k; ++i) {
            const auto rc = ro[i];
            res[i][invCo[rc]] = rc;
        }

        return res;
    }
};
58467