knightnemo/LMCompressionDataset · Datasets at Hugging Face

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For example, if $n=5$, $f=2$, $a = [4, \color{green}3, 3, 2, 3]$ (the favorite cube is highlighted in green), and $k = 2$, the following could have happened:

* After sorting $a=[4, \color{green}3, 3, 3, 2]$, since the favorite cube ended up in the second position, it will be removed. * After sorting $a=[4, 3, \color{green}3, 3, 2]$, since the favorite cube ended up in the third position, it will not be removed.

The first line contains an integer $t$ ($1 \le t \le 1000$) — the number of test cases. Then follow the descriptions of the test cases.

The first line of each test case description contains three integers $n$, $f$, and $k$ ($1 \le f, k \le n \le 100$) — the number of cubes, the index of Dmitry's favorite cube, and the number of removed cubes, respectively.

The second line of each test case description contains $n$ integers $a_i$ ($1 \le a_i \le 100$) — the values shown on the cubes.

For each test case, output one line — "YES" if the cube will be removed in all cases, "NO" if it will not be removed in any case, "MAYBE" if it may be either removed or left.

You can output the answer in any case. For example, the strings "YES", "nO", "mAyBe" will be accepted as answers.

Sofia had an array of $n$ integers $a_1, a_2, \ldots, a_n$. One day she got bored with it, so she decided to sequentially apply $m$ modification operations to it.

Each modification operation is described by a pair of numbers $\langle c_j, d_j \rangle$ and means that the element of the array with index $c_j$ should be assigned the value $d_j$, i.e., perform the assignment $a_{c_j} = d_j$. After applying all modification operations sequentially, Sofia discarded the resulting array.

Recently, you found an array of $n$ integers $b_1, b_2, \ldots, b_n$. You are interested in whether this array is Sofia's array. You know the values of the original array, as well as the values $d_1, d_2, \ldots, d_m$. The values $c_1, c_2, \ldots, c_m$ turned out to be lost.

Is there a sequence $c_1, c_2, \ldots, c_m$ such that the sequential application of modification operations $\langle c_1, d_1, \rangle, \langle c_2, d_2, \rangle, \ldots, \langle c_m, d_m \rangle$ to the array $a_1, a_2, \ldots, a_n$ transforms it into the array $b_1, b_2, \ldots, b_n$?

The first line contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Then follow the descriptions of the test cases.

The first line of each test case contains an integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the original array.

The third 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 found array.

The fourth line contains an integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of modification operations.

The fifth line contains $m$ integers $d_1, d_2, \ldots, d_m$ ($1 \le d_j \le 10^9$) — the preserved value for each modification operation.

It is guaranteed that the sum of the values of $n$ for all test cases does not exceed $2 \cdot 10^5$, similarly the sum of the values of $m$ for all test cases does not exceed $2 \cdot 10^5$.

Output $
You have been given a matrix $a$ of size $n$ by $m$, containing a permutation of integers from $1$ to $n \cdot m$.

A permutation of $n$ integers is an array containing all numbers from $1$ to $n$ exactly once. For example, the arrays $[1]$, $[2, 1, 3]$, $[5, 4, 3, 2, 1]$ are permutations, while the arrays $[1, 1]$, $[100]$, $[1, 2, 4, 5]$ are not.

A matrix contains a permutation if, when all its elements are written out, the resulting array is a permutation. Matrices $[[1, 2], [3, 4]]$, $[[1]]$, $[[1, 5, 3], [2, 6, 4]]$ contain permutations, while matrices $[[2]]$, $[[1, 1], [2, 2]]$, $[[1, 2], [100, 200]]$ do not.

You can perform one of the following two actions in one operation:

* choose columns $c$ and $d$ ($1 \le c, d \le m$, $c \ne d$) and swap these columns; * choose rows $c$ and $d$ ($1 \le c, d \le n$, $c \ne d$) and swap these rows.

You can perform any number of operations.

You are given the original matrix $a$ and the matrix $b$. Your task is to determine whether it is possible to transform matrix $a$ into matrix $b$ using the given operations.

The first line contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases. The descriptions of the test cases follow.

The first line of each test case description contains $2$ integers $n$ and $m$ ($1 \le n, m \le n \cdot m \le 2 \cdot 10^5$) — the sizes of the matrix.

The next $n$ lines contain $m$ integers $a_{ij}$ each ($1 \le a_{ij} \le n \cdot m$). It is guaranteed that matrix $a$ is a permutation.

The next $n$ lines contain $m$ integers $b_{ij}$ each ($1 \le b_{ij} \le n \cdot m$). It is guaranteed that matrix $b$ is a permutation.

It is guaranteed that the sum of the values $n \cdot m$ for all test cases does not exceed $2 \cdot 10^5$.

For each test case, output "YES" if the second matrix can be obtained from the first, and "NO" otherwise.

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

In the second example,
There is a sequence $a_0, a_1, a_2, \ldots$ of infinite length. Initially $a_i = i$ for every non-negative integer $i$.

After every second, each element of the sequence will simultaneously change. $a_i$ will change to $a_{i - 1} \mid a_i \mid a_{i + 1}$ for every positive integer $i$. $a_0$ will change to $a_0 \mid a_1$. Here, $\|$ denotes [bitwise OR](https://en.wikipedia.org/wiki/Bitwise_operation#OR).

Turtle is asked to find the value of $a_n$ after $m$ seconds. In particular, if $m = 0$, then he needs to find the initial value of $a_n$. He is tired of calculating so many values, so please help him!

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 two integers $n, m$ ($0 \le n, m \le 10^9$).

For each test case, output a single integer — the value of $a_n$ after $m$ seconds.

After $1$ second, $[a_0, a_1, a_2, a_3, a_4, a_5]$ will become $[1, 3, 3, 7, 7, 7]$.

After $2$ seconds, $[a_0, a_1, a_2, a_3, a_4, a_5]$ will become $[3, 3, 7, 7, 7, 7]$.
Turtle just received $n$ segments and a sequence $a_1, a_2, \ldots, a_n$. The $i$-th segment is $[l_i, r_i]$.

Turtle will create an undirected graph $G$. If segment $i$ and segment $j$ intersect, then Turtle will add an undirected edge between $i$ and $j$ with a weight of $\|a_i - a_j\|$, for every $i \ne j$.

Turtle wants you to calculate the sum of the weights of the edges of the minimum spanning tree of the graph $G$, or report that the graph $G$ has no spanning tree.

We say two segments $[l_1, r_1]$ and $[l_2, r_2]$ intersect if and only if $\max(l_1, l_2) \le \min(r_1, r_2)$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 5 \cdot 10^5$) — the number of segments.

The $i$-th of the following $n$ lines contains three integers $l_i, r_i, a_i$ ($1 \le l_i \le r_i \le 10^9, 1 \le a_i \le 10^9$) — the $i$-th segment and the $i$-th element of the sequence.

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

For example, if $n=5$, $f=2$, $a = [4, \color{green}3, 3, 2, 3]$ (the favorite cube is highlighted in green), and $k = 2$, the following could have happened:

* After sorting $a=[4, \color{green}3, 3, 3, 2]$, since the favorite cube ended up in the second position, it will be removed. * After sorting $a=[4, 3, \color{green}3, 3, 2]$, since the favorite cube ended up in the third position, it will not be removed.

The first line contains an integer $t$ ($1 \le t \le 1000$) — the number of test cases. Then follow the descriptions of the test cases.

The first line of each test case description contains three integers $n$, $f$, and $k$ ($1 \le f, k \le n \le 100$) — the number of cubes, the index of Dmitry's favorite cube, and the number of removed cubes, respectively.

The second line of each test case description contains $n$ integers $a_i$ ($1 \le a_i \le 100$) — the values shown on the cubes.

For each test case, output one line — "YES" if the cube will be removed in all cases, "NO" if it will not be removed in any case, "MAYBE" if it may be either removed or left.

You can output the answer in any case. For example, the strings "YES", "nO", "mAyBe" will be accepted as answers.

Sofia had an array of $n$ integers $a_1, a_2, \ldots, a_n$. One day she got bored with it, so she decided to sequentially apply $m$ modification operations to it.

Each modification operation is described by a pair of numbers $\langle c_j, d_j \rangle$ and means that the element of the array with index $c_j$ should be assigned the value $d_j$, i.e., perform the assignment $a_{c_j} = d_j$. After applying all modification operations sequentially, Sofia discarded the resulting array.

Recently, you found an array of $n$ integers $b_1, b_2, \ldots, b_n$. You are interested in whether this array is Sofia's array. You know the values of the original array, as well as the values $d_1, d_2, \ldots, d_m$. The values $c_1, c_2, \ldots, c_m$ turned out to be lost.

Is there a sequence $c_1, c_2, \ldots, c_m$ such that the sequential application of modification operations $\langle c_1, d_1, \rangle, \langle c_2, d_2, \rangle, \ldots, \langle c_m, d_m \rangle$ to the array $a_1, a_2, \ldots, a_n$ transforms it into the array $b_1, b_2, \ldots, b_n$?

The first line contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Then follow the descriptions of the test cases.

The first line of each test case contains an integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the original array.

The third 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 found array.

The fourth line contains an integer $m$ ($1 \le m \le 2 \cdot 10^5$) — the number of modification operations.

The fifth line contains $m$ integers $d_1, d_2, \ldots, d_m$ ($1 \le d_j \le 10^9$) — the preserved value for each modification operation.

It is guaranteed that the sum of the values of $n$ for all test cases does not exceed $2 \cdot 10^5$, similarly the sum of the values of $m$ for all test cases does not exceed $2 \cdot 10^5$.

Output $

You have been given a matrix $a$ of size $n$ by $m$, containing a permutation of integers from $1$ to $n \cdot m$.

A permutation of $n$ integers is an array containing all numbers from $1$ to $n$ exactly once. For example, the arrays $[1]$, $[2, 1, 3]$, $[5, 4, 3, 2, 1]$ are permutations, while the arrays $[1, 1]$, $[100]$, $[1, 2, 4, 5]$ are not.

A matrix contains a permutation if, when all its elements are written out, the resulting array is a permutation. Matrices $[[1, 2], [3, 4]]$, $[[1]]$, $[[1, 5, 3], [2, 6, 4]]$ contain permutations, while matrices $[[2]]$, $[[1, 1], [2, 2]]$, $[[1, 2], [100, 200]]$ do not.

You can perform one of the following two actions in one operation:

* choose columns $c$ and $d$ ($1 \le c, d \le m$, $c \ne d$) and swap these columns; * choose rows $c$ and $d$ ($1 \le c, d \le n$, $c \ne d$) and swap these rows.

You can perform any number of operations.

You are given the original matrix $a$ and the matrix $b$. Your task is to determine whether it is possible to transform matrix $a$ into matrix $b$ using the given operations.

The first line contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases. The descriptions of the test cases follow.

The first line of each test case description contains $2$ integers $n$ and $m$ ($1 \le n, m \le n \cdot m \le 2 \cdot 10^5$) — the sizes of the matrix.

The next $n$ lines contain $m$ integers $a_{ij}$ each ($1 \le a_{ij} \le n \cdot m$). It is guaranteed that matrix $a$ is a permutation.

The next $n$ lines contain $m$ integers $b_{ij}$ each ($1 \le b_{ij} \le n \cdot m$). It is guaranteed that matrix $b$ is a permutation.

It is guaranteed that the sum of the values $n \cdot m$ for all test cases does not exceed $2 \cdot 10^5$.

For each test case, output "YES" if the second matrix can be obtained from the first, and "NO" otherwise.

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

In the second example,

There is a sequence $a_0, a_1, a_2, \ldots$ of infinite length. Initially $a_i = i$ for every non-negative integer $i$.

After every second, each element of the sequence will simultaneously change. $a_i$ will change to $a_{i - 1} \mid a_i \mid a_{i + 1}$ for every positive integer $i$. $a_0$ will change to $a_0 \mid a_1$. Here, $|$ denotes [bitwise OR](https://en.wikipedia.org/wiki/Bitwise_operation#OR).

Turtle is asked to find the value of $a_n$ after $m$ seconds. In particular, if $m = 0$, then he needs to find the initial value of $a_n$. He is tired of calculating so many values, so please help him!

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 two integers $n, m$ ($0 \le n, m \le 10^9$).

For each test case, output a single integer — the value of $a_n$ after $m$ seconds.

After $1$ second, $[a_0, a_1, a_2, a_3, a_4, a_5]$ will become $[1, 3, 3, 7, 7, 7]$.

After $2$ seconds, $[a_0, a_1, a_2, a_3, a_4, a_5]$ will become $[3, 3, 7, 7, 7, 7]$.

Turtle just received $n$ segments and a sequence $a_1, a_2, \ldots, a_n$. The $i$-th segment is $[l_i, r_i]$.

Turtle will create an undirected graph $G$. If segment $i$ and segment $j$ intersect, then Turtle will add an undirected edge between $i$ and $j$ with a weight of $|a_i - a_j|$, for every $i \ne j$.

Turtle wants you to calculate the sum of the weights of the edges of the minimum spanning tree of the graph $G$, or report that the graph $G$ has no spanning tree.

We say two segments $[l_1, r_1]$ and $[l_2, r_2]$ intersect if and only if $\max(l_1, l_2) \le \min(r_1, r_2)$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 5 \cdot 10^5$) — the number of segments.

The $i$-th of the following $n$ lines contains three integers $l_i, r_i, a_i$ ($1 \le l_i \le r_i \le 10^9, 1 \le a_i \le 10^9$) — the $i$-th segment and the $i$-th element of the sequence.

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