Permutation Swap

1000ms

262144K

Description:

You are given an unsorted permutation $$$p_1, p_2, \ldots, p_n$$$. To sort the permutation, you choose a constant $$$k$$$ ($$$k \ge 1$$$) and do some operations on the permutation. In one operation, you can choose two integers $$$i$$$, $$$j$$$ ($$$1 \le j < i \le n$$$) such that $$$i - j = k$$$, then swap $$$p_i$$$ and $$$p_j$$$.

What is the maximum value of $$$k$$$ that you can choose to sort the given permutation?

A permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2, 3, 1, 5, 4]$$$ is a permutation, but $$$[1, 2, 2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1, 3, 4]$$$ is also not a permutation ($$$n = 3$$$ but there is $$$4$$$ in the array).

An unsorted permutation $$$p$$$ is a permutation such that there is at least one position $$$i$$$ that satisfies $$$p_i \ne i$$$.

Input:

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 a single integer $$$n$$$ ($$$2 \le n \le 10^{5}$$$) — the length of the permutation $$$p$$$.

The second line of each test case contains $$$n$$$ distinct integers $$$p_1, p_2, \ldots, p_n$$$ ($$$1 \le p_i \le n$$$) — the permutation $$$p$$$. It is guaranteed that the given numbers form a permutation of length $$$n$$$ and the given permutation is unsorted.

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

Output:

For each test case, output the maximum value of $$$k$$$ that you can choose to sort the given permutation.

We can show that an answer always exists.

Sample Input:

7
3
3 1 2
4
3 4 1 2
7
4 2 6 7 5 3 1
9
1 6 7 4 9 2 3 8 5
6
1 5 3 4 2 6
10
3 10 5 2 9 6 7 8 1 4
11
1 11 6 4 8 3 7 5 9 10 2

Sample Output:

Note:

In the first test case, the maximum value of $$$k$$$ you can choose is $$$1$$$. The operations used to sort the permutation are:

Swap $$$p_2$$$ and $$$p_1$$$ ($$$2 - 1 = 1$$$) $$$\rightarrow$$$ $$$p = [1, 3, 2]$$$
Swap $$$p_2$$$ and $$$p_3$$$ ($$$3 - 2 = 1$$$) $$$\rightarrow$$$ $$$p = [1, 2, 3]$$$

In the second test case, the maximum value of $$$k$$$ you can choose is $$$2$$$. The operations used to sort the permutation are:

Swap $$$p_3$$$ and $$$p_1$$$ ($$$3 - 1 = 2$$$) $$$\rightarrow$$$ $$$p = [1, 4, 3, 2]$$$
Swap $$$p_4$$$ and $$$p_2$$$ ($$$4 - 2 = 2$$$) $$$\rightarrow$$$ $$$p = [1, 2, 3, 4]$$$

Informação

Provedor Codeforces

Código CF1828B