A15648. Lucky Permutation
编程题
普及/提高-
知识点
题目描述
You are given a permutation $^\dagger$ $p$ of length $n$ .
In one operation, you can choose two indices $1 \le i < j \le n$ and swap $p_i$ with $p_j$ .
Find the minimum number of operations needed to have exactly one inversion $^\ddagger$ in the permutation.
$^\dagger$ 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).
$^\ddagger$ The number of inversions of a permutation $p$ is the number of pairs of indices $(i, j)$ such that $1 \le i < j \le n$ and $p_i > p_j$ .
In one operation, you can choose two indices $1 \le i < j \le n$ and swap $p_i$ with $p_j$ .
Find the minimum number of operations needed to have exactly one inversion $^\ddagger$ in the permutation.
$^\dagger$ 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).
$^\ddagger$ The number of inversions of a permutation $p$ is the number of pairs of indices $(i, j)$ such that $1 \le i < j \le n$ and $p_i > p_j$ .
输入格式
The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. The description of test cases follows.
The first line of each test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).
The second line of each test case contains $n$ integers $p_1,p_2,\ldots, p_n$ ( $1 \le p_i \le n$ ). It is guaranteed that $p$ is a permutation.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .
The first line of each test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).
The second line of each test case contains $n$ integers $p_1,p_2,\ldots, p_n$ ( $1 \le p_i \le n$ ). It is guaranteed that $p$ is a permutation.
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 a single integer — the minimum number of operations needed to have exactly one inversion in the permutation. It can be proven that an answer always exists.
输入输出样例
输入 #1
4 2 2 1 2 1 2 4 3 4 1 2 4 2 4 3 1
输出 #1
0 1 3 1
说明/提示
In the first test case, the permutation already satisfies the condition.
In the second test case, you can perform the operation with $(i,j)=(1,2)$ , after that the permutation will be $[2,1]$ which has exactly one inversion.
In the third test case, it is not possible to satisfy the condition with less than $3$ operations. However, if we perform $3$ operations with $(i,j)$ being $(1,3)$ , $(2,4)$ , and $(3,4)$ in that order, the final permutation will be $[1, 2, 4, 3]$ which has exactly one inversion.
In the fourth test case, you can perform the operation with $(i,j)=(2,4)$ , after that the permutation will be $[2,1,3,4]$ which has exactly one inversion.
In the second test case, you can perform the operation with $(i,j)=(1,2)$ , after that the permutation will be $[2,1]$ which has exactly one inversion.
In the third test case, it is not possible to satisfy the condition with less than $3$ operations. However, if we perform $3$ operations with $(i,j)$ being $(1,3)$ , $(2,4)$ , and $(3,4)$ in that order, the final permutation will be $[1, 2, 4, 3]$ which has exactly one inversion.
In the fourth test case, you can perform the operation with $(i,j)=(2,4)$ , after that the permutation will be $[2,1,3,4]$ which has exactly one inversion.