310815: CF1893A. 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
给定一个数组 $ b_1, b_2, \ldots, b_n $。一个匿名线人告诉你，这个数组 $ b $ 是通过以下方式获得的：最初存在一个数组 $ a_1, a_2, \ldots, a_n $，然后执行了 $ k $ 次以下两个步骤的操作：
1. 选择数组 $ a $ 的一个固定点 $ x $。
2. 然后，数组 $ a $ 向左循环移动了 $ x $ 次。
经过 $ k $ 次这样的操作后，得到了数组 $ b_1, b_2, \ldots, b_n $。你需要检查匿名线人的话是否可能为真，或者是否一定为假。
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。
- 每个测试用例包含两行：
- 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为真，输出 "Yes"，如果一定为假，输出 "No"。题目大意： 给定一个数组 $ b_1, b_2, \ldots, b_n $。一个匿名线人告诉你，这个数组 $ b $ 是通过以下方式获得的：最初存在一个数组 $ a_1, a_2, \ldots, a_n $，然后执行了 $ k $ 次以下两个步骤的操作： 1. 选择数组 $ a $ 的一个固定点 $ x $。 2. 然后，数组 $ a $ 向左循环移动了 $ x $ 次。 经过 $ k $ 次这样的操作后，得到了数组 $ b_1, b_2, \ldots, b_n $。你需要检查匿名线人的话是否可能为真，或者是否一定为假。 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。 - 每个测试用例包含两行： - 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为真，输出 "Yes"，如果一定为假，输出 "No"。