Sort with Step
Contest: Codeforces Round 868 (Div. 2)
Difficulty rating: 900
Tags: brute force, math, sortings

Let's define a permutation of length $$$n$$$ as an array $$$p$$$ of length $$$n$$$, which contains every number from $$$1$$$ to $$$n$$$ exactly once.

You are given a permutation $$$p_1, p_2, \dots, p_n$$$ and a number $$$k$$$. You need to sort this permutation in the ascending order. In order to do it, you can repeat the following operation any number of times (possibly, zero):

- pick two elements of the permutation $$$p_i$$$ and $$$p_j$$$ such that $$$|i - j| = k$$$, and swap them.

Unfortunately, some permutations can't be sorted with some fixed numbers $$$k$$$. For example, it's impossible to sort $$$[2, 4, 3, 1]$$$ with $$$k = 2$$$.

That's why, before starting the sorting, you can make at most one preliminary exchange:

- choose any pair $$$p_i$$$ and $$$p_j$$$ and swap them.

Your task is to:

1. check whether is it possible to sort the permutation without any preliminary exchanges,
2. if it's not, check, whether is it possible to sort the permutation using exactly one preliminary exchange.

For example, if $$$k = 2$$$ and permutation is $$$[2, 4, 3, 1]$$$, then you can make a preliminary exchange of $$$p_1$$$ and $$$p_4$$$, which will produce permutation $$$[1, 4, 3, 2]$$$, which is possible to sort with given $$$k$$$.

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

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \le n \le 2 \cdot 10^5$$$; $$$1 \le k \le n - 1$$$) — length of the permutation, and a distance between elements that can be swapped.

The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, \dots, p_n$$$ ($$$1 \le p_i \le n$$$) — elements of the permutation $$$p$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10 ^ 5$$$.

Output format:
For each test case print

- 0, if it is possible to sort the permutation without preliminary exchange;
- 1, if it is possible to sort the permutation with one preliminary exchange, but not possible without preliminary exchange;
- -1, if it is not possible to sort the permutation with at most one preliminary exchange.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first test case, there is no need in preliminary exchange, as it is possible to swap $$$(p_1, p_2)$$$ and then $$$(p_2, p_3)$$$.

In the second test case, there is no need in preliminary exchange, as it is possible to swap $$$(p_1, p_3)$$$ and then $$$(p_2, p_4)$$$.

In the third test case, you need to apply preliminary exchange to $$$(p_2, p_3)$$$. After that the permutation becomes $$$[3, 4, 1, 2]$$$ and can be sorted with $$$k = 2$$$.

Example input:
6
4 1
3 1 2 4
4 2
3 4 1 2
4 2
3 1 4 2
10 3
4 5 9 1 8 6 10 2 3 7
10 3
4 6 9 1 8 5 10 2 3 7
10 3
4 6 9 1 8 5 10 3 2 7

Example output:
0
0
1
0
1
-1

Editorial:
Let's fix a number $$$0 \le x < k$$$ and consider all indices $$$i$$$ such that $$$i \bmod k = x$$$. Note that numbers in these positions can be reordered however we want, but they can't leave this set of positions, since $$$i \bmod k$$$ stay the same. Moreover, in the sorted permutation these positions must contain numbers $$$j$$$ such that $$$(j - 1) \bmod k = x$$$. In total, itm means that the permutation can be sorted if $$$(p[i] - 1) \bmod k = i \bmod k$$$ holds for all $$$i$$$.
By preliminary exchange, we can swap two elements from different sets. Therefore, if the last equality fails for exactly two elements, they can be swapped, making sorting possible. Otherwise, the answer is $$$-1$$$.