310822: CF1894C. Anonymous Informant
Description
You are given an array $b_1, b_2, \ldots, b_n$.
An anonymous informant has told you that the array $b$ was obtained as follows: initially, there existed an array $a_1, a_2, \ldots, a_n$, after which the following two-component operation was performed $k$ times:
- A fixed point$^{\dagger}$ $x$ of the array $a$ was chosen.
- Then, the array $a$ was cyclically shifted to the left$^{\ddagger}$ exactly $x$ times.
As a result of $k$ such operations, the array $b_1, b_2, \ldots, b_n$ was obtained. You want to check if the words of the anonymous informant can be true or if they are guaranteed to be false.
$^{\dagger}$A number $x$ is called a fixed point of the array $a_1, a_2, \ldots, a_n$ if $1 \leq x \leq n$ and $a_x = x$.
$^{\ddagger}$A cyclic left shift of the array $a_1, a_2, \ldots, a_n$ is the array $a_2, \ldots, a_n, a_1$.
InputEach 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 $n, k$ ($1 \le n \le 2 \cdot 10^5$, $1 \le k \le 10^9$) — the length of the array $b$ and the number of operations performed.
The second line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le 10^9$) — the elements of the array $b$.
It is guaranteed that the sum of the values of $n$ for all test cases does not exceed $2 \cdot 10^5$.
OutputFor each test case, output "Yes" if the words of the anonymous informant can be true, and "No" if they are guaranteed to be false.
ExampleInput6 5 3 4 3 3 2 3 3 100 7 2 1 5 5 6 1 1 1 1 1 1000000000 1 8 48 9 10 11 12 13 14 15 8 2 1 1 42Output
Yes Yes No Yes Yes NoNote
In the first test case, the array $a$ could be equal to $[3, 2, 3, 4, 3]$. In the first operation, a fixed point $x = 2$ was chosen, and after $2$ left shifts, the array became $[3, 4, 3, 3, 2]$. In the second operation, a fixed point $x = 3$ was chosen, and after $3$ left shifts, the array became $[3, 2, 3, 4, 3]$. In the third operation, a fixed point $x = 3$ was chosen again, and after $3$ left shifts, the array became $[4, 3, 3, 2, 3]$, which is equal to the array $b$.
In the second test case, the array $a$ could be equal to $[7, 2, 1]$. After the operation with a fixed point $x = 2$, the array became $[1, 7, 2]$. Then, after the operation with a fixed point $x = 1$, the array returned to its initial state $[7, 2, 1]$. These same $2$ operations (with $x = 2$, and $x = 1$) were repeated $49$ times. So, after $100$ operations, the array returned to $[7, 2, 1]$.
In the third test case, it can be shown that there is no solution.
Output
你有一个数组$b_1, b_2, \ldots, b_n$。一个匿名线人告诉你，这个数组$b$是通过以下方式得到的：最初存在一个数组$a_1, a_2, \ldots, a_n$，然后执行了以下两个步骤的操作$k$次：
1. 选择数组$a$的一个固定点$x$。
2. 然后，数组$a$向左循环移动$x$次。
经过$k$次这样的操作后，得到了数组$b_1, b_2, \ldots, b_n$。你想要检查匿名线人的话是否可能是真的，或者是否一定是假的。
输入输出数据格式：
输入：
- 第一行包含一个整数$t$ ($1 \le t \le 10^4$) —— 测试用例的数量。
- 每个测试用例包含以下内容：
- 第一行包含两个整数$n, k$ ($1 \le n \le 2 \cdot 10^5$, $1 \le k \le 10^9$) —— 数组$b$的长度和执行的 operations。
- 第二行包含$n$个整数$b_1, b_2, \ldots, b_n$ ($1 \le b_i \le 10^9$) —— 数组$b$的元素。
输出：
- 对于每个测试用例，如果匿名线人的话可能是真的，输出"Yes"，如果一定是不真的，输出"No"。题目大意： 你有一个数组$b_1, b_2, \ldots, b_n$。一个匿名线人告诉你，这个数组$b$是通过以下方式得到的：最初存在一个数组$a_1, a_2, \ldots, a_n$，然后执行了以下两个步骤的操作$k$次： 1. 选择数组$a$的一个固定点$x$。 2. 然后，数组$a$向左循环移动$x$次。 经过$k$次这样的操作后，得到了数组$b_1, b_2, \ldots, b_n$。你想要检查匿名线人的话是否可能是真的，或者是否一定是假的。 输入输出数据格式： 输入： - 第一行包含一个整数$t$ ($1 \le t \le 10^4$) —— 测试用例的数量。 - 每个测试用例包含以下内容： - 第一行包含两个整数$n, k$ ($1 \le n \le 2 \cdot 10^5$, $1 \le k \le 10^9$) —— 数组$b$的长度和执行的 operations。 - 第二行包含$n$个整数$b_1, b_2, \ldots, b_n$ ($1 \le b_i \le 10^9$) —— 数组$b$的元素。 输出： - 对于每个测试用例，如果匿名线人的话可能是真的，输出"Yes"，如果一定是不真的，输出"No"。