2392 - Build a Matrix With Conditions.

Hard

You are given a positive integer k. You are also given:

• a 2D integer array rowConditions of size n where <code>rowConditionsi = above_i, below_i</code>, and

• a 2D integer array colConditions of size m where <code>colConditionsi = left_i, right_i</code>.

The two arrays contain integers from 1 to k.

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

The matrix should also satisfy the following conditions:

• The number <code>above_i</code> should appear in a row that is strictly above the row at which the number <code>below_i</code> appears for all i from 0 to n - 1.

• The number <code>left_i</code> should appear in a column that is strictly left of the column at which the number <code>right_i</code> appears for all i from 0 to m - 1.

Return any matrix that satisfies the conditions. If no answer exists, return an empty matrix.

Example 1:

Input: k = 3, rowConditions = \[\[1,2],3,2], colConditions = \[\[2,1],3,2]

Output: [3,0,0,0,0,1,0,2,0]

Explanation: 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 <ins>1</ins>, and number 2 is in row <ins>2</ins>, so 1 is above 2 in the matrix.

• Number 3 is in row <ins>0</ins>, and number 2 is in row <ins>2</ins>, so 3 is above 2 in the matrix.

The column conditions are the following:

• Number 2 is in column <ins>1</ins>, and number 1 is in column <ins>2</ins>, so 2 is left of 1 in the matrix.

• Number 3 is in column <ins>0</ins>, and number 2 is in column <ins>1</ins>, so 3 is left of 2 in the matrix.

Note that there may be multiple correct answers.

Example 2:

Input: k = 3, rowConditions = \[\[1,2],2,3,3,1,2,3], colConditions = \[\[2,1]]

Output: []

Explanation: 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.

Constraints:

• 2 <= k <= 400

• <code>1 <= rowConditions.length, colConditions.length <= 10⁴</code>

• rowConditions[i].length == colConditions[i].length == 2

• <code>1 <= above_i, below_i, left_i, right_i<= k</code>

• <code>above_i != below_i</code>

• <code>left_i != right_i</code>