F. Inversion Composition
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
My Chemical Romance - Disenchanted
ඞ

You are given a permutation $$$p$$$ of size $$$n$$$, as well as a non-negative integer $$$k$$$. You need to construct a permutation $$$q$$$ of size $$$n$$$ such that $$$\operatorname{inv}(q) + \operatorname{inv}(q \cdot p) = k {}^\dagger {}^\ddagger$$$, or determine if it is impossible to do so.

$$${}^\dagger$$$ For two permutations $$$p$$$ and $$$q$$$ of the same size $$$n$$$, the permutation $$$w = q \cdot p$$$ is such that $$$w_i = q_{p_i}$$$ for all $$$1 \le i \le n$$$.

$$${}^\ddagger$$$ For a permutation $$$p$$$ of size $$$n$$$, the function $$$\operatorname{inv}(p)$$$ returns the number of inversions of $$$p$$$, i.e. the number of pairs of indices $$$1 \le i < j \le n$$$ such that $$$p_i > p_j$$$.

Input

Each test contains multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 3 \cdot 10^5, 0 \le k \le n(n - 1)$$$) — the size of $$$p$$$ and the target number of inversions.

The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, \ldots, p_n$$$ ($$$1 \le p_i \le n$$$, $$$p_i$$$'s are pairwise distinct) — the given permutation $$$p$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output

For each test case, print on one line "YES" if there exists a permutation $$$q$$$ that satisfies the given condition, or "NO" if there is no such permutation.

If the answer is "YES", on the second line, print $$$n$$$ integers $$$q_1, q_2, \ldots, q_n$$$ that represent such a satisfactory permutation $$$q$$$. If there are multiple such $$$q$$$'s, print any of them.

Example
Input
5
3 4
2 3 1
5 5
2 3 5 1 4
6 11
5 1 2 3 4 6
9 51
3 1 4 2 5 6 7 8 9
1 0
1
Output
YES
3 2 1
NO
NO
YES
1 5 9 8 7 6 4 3 2
YES
1
Note

In the first test case, we have $$$q \cdot p = [2, 1, 3]$$$, $$$\operatorname{inv}(q) = 3$$$, and $$$\operatorname{inv}(q \cdot p) = 1$$$.

In the fourth test case, we have $$$q \cdot p = [9, 1, 8, 5, 7, 6, 4, 3, 2]$$$, $$$\operatorname{inv}(q) = 24$$$, and $$$\operatorname{inv}(q \cdot p) = 27$$$.