J. Permutation Puzzle
time limit per test
4 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

Little relyt871 is solving a puzzle. The key to the puzzle is a permutation containing numbers $$$1 \dots n$$$. The values at some positions of the permutation are already fixed, and relyt871 needs to fill numbers into the remaining positions.

Besides, little relyt871 has gathered $$$m$$$ extra requirements about the permutation. Let the solution be represented as $$$p_1,p_2,...,p_n$$$, each clue is a pair of indices $$$(u_i,v_i)$$$, which means that $$$p_{u_i} \lt p_{v_i}$$$ should be satisfied in the solution.

Little relyt871 wonders if all requirements may be satisfied at the same time. Write a program to find a valid solution when there is one.

Input

The first line of the input contains the number of test cases $$$T\ (1 \leq T \leq 20\,000)$$$.

For each test case:

  • The first line contains two integers $$$n,m\ (2 \leq n \leq 200\,000, 1 \leq m \leq 500\,000)$$$.
  • The second line contains $$$n$$$ integers $$$p_1,p_2,...,p_n\ (0 \leq p_i \leq n)$$$. If $$$1 \leq p_i \leq n$$$, then the value at position $$$i$$$ is fixed as $$$p_i$$$, otherwise it is your task to determine the value at position $$$i$$$. It is guaranteed that for $$$1 \leq i \lt j \leq n$$$, if $$$p_i \gt 0$$$ and $$$p_j \gt 0$$$, then $$$p_i \neq p_j$$$.
  • The following $$$m$$$ lines each contains two integers $$$u_i,v_i\ (1 \leq u_i,v_i \leq n)$$$, denoting the clues. It is guaranteed that the clues don't contradict themselves. Formally, there doesn't exist a list of clues $$$(u_{i_1},v_{i_1}), (u_{i_2},v_{i_2}),...,(u_{i_k},v_{i_k})$$$ such that $$$v_{i_j} = u_{i_{j+1}},1\leq j \lt k$$$ and $$$v_{i_k} = u_{i_1}$$$.

The sum of $$$n$$$ over all test cases doesn't exceed $$$200\,000$$$, and the sum of $$$m$$$ doesn't exceed $$$500\,000$$$.

Output

For each test case:

  • If there exists no valid solution, output "-1" in a single line.
  • Otherwise, output one line containing $$$n$$$ integers seperated by spaces, denoting the solution. If there are multiple solutions, print any.
Examples
Input
2
4 4
1 0 0 4
1 2
1 3
2 4
3 4
3 2
0 3 1
1 2
3 1
Output
1 3 2 4 
2 3 1
Input
1
4 4
1 4 0 0
1 2
1 3
2 4
3 4
Output
-1