Permutation Swap

题面翻译

给你一个长度为 $n$ 的未排序的排列。找到最大的整数 $k$ 满足可以通过只交换下标差为 $k$ 的元素使排列被从小到大排序。

题目描述

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 $ .

输入格式

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} $ .

输出格式

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.

样例 #1

样例输入 #1

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

样例输出 #1

1
2
3
4
3
2
3

提示

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] $

并不是找离正确位置最小的那一个，因为最小的那个k不一定会让其他的也可以回到正确位置，所以要找的是最大公约数

int gcd(int a, int b)
{
    return b ? gcd(b, a % b) : a;
}
void solve()
{
    int n; cin >> n;
    int g;
    for (int i = 1; i <= n; i++)
    {
        int x; cin >> x;
        if (i == 1) g = abs(x - i);
        else g = gcd(g, abs(x - i));
    }
    cout << g << '\n';
}