knightnemo/LMCompressionDataset · Datasets at Hugging Face

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For each test case, output a single integer — the sum of the weights of the edges of the minimum spanning tree of the graph $G$. If the graph $G$ has no spanning tree, output $-1$.

In the first test case, the graph $G$ is as follows:

![](CDN_BASE_URL/7b0a32f8f1d54a5dbbaffee1efd8ece1)

One of the minimum spanning trees of $G$ is as follows:

![](CDN_BASE_URL/7a035af2d5227648781d0c24b81c076e)

The sum of the weights of the edges of the minimum spanning tree is $9$.

In the second test case, the graph $G$ is as follows:

![](CDN_BASE_URL/62be880a08fa85071b491d6badd5f58e)

$G$ is already a tree, and the sum of the weights of the tree is $13$.

In the third test case, the graph $G$ is as follows:

![](CDN_BASE_URL/b667aa771d019e3a8a3a4c4372200f03)

In the fourth test case, the graph $G$ is as follows:

![](CDN_BASE_URL/45bc2d052677dbf032ff37fd1d723d83)

It's easy to see that $G$ is no
On another boring day, Egor got bored and decided to do something. But since he has no friends, he came up with a game to play.

Egor has a deck of $n$ cards, the $i$-th card from the top has a number $a_i$ written on it. Egor wants to play a certain number of rounds until the cards run out. In each round, he takes a non-zero number of cards from the top of the deck and finishes the round. If the sum of the numbers on the cards collected during the round is between $l$ and $r$, inclusive, the round is won; otherwise, it is lost.

Egor knows by heart the order of the cards. Help Egor determine the maximum number of rounds he can win in such a game. Note that Egor is not required to win rounds consecutively.

Each test consists of several test cases. The first line contains an integer $t$ ($1 \le t \le 10^{4}$) — the number of test cases. This is followed by a description of the test cases.

The first line of each test case contains three integers $n$, $l$, and $r$ ($1 \le n \le 10^{5}$, $1 \le l \le r \le 10^9$).

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 numbers on the cards from top to bottom.

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

For each test case, output a single number — the maximum number of rounds Egor can win.

In the first test case, Egor can win $3$ rounds:

* In the first round, take the top $2$ cards with values $2$ and $1$ and win, as their sum is $3$. After this, the deck will look like this: $[11, 3, 7]$. * In the second round, take the top card and lose, as its value $11$ is greater than $r = 10$. After this, the deck will look like this: $[3, 7]$. * In the third round, take the top card with value $3$ and win. After this, the deck will look like this: $[7]$. * After this, in the fourth round, Egor only has to take the last card in the deck with value $7$ and win again.

In the second test case, Egor cannot win any rounds, no matter how hard he tries.

In the t
Given two arrays of distinct positive integers $a$ and $b$ of length $n$, we would like to make both the arrays the same. Two arrays $x$ and $y$ of length $k$ are said to be the same when for all $1 \le i \le k$, $x_i = y_i$.

Now in one move, you can choose some index $l$ and $r$ in $a$ ($l \le r$) and swap $a_l$ and $a_r$, then choose some $p$ and $q$ ($p \le q$) in $b$ such that $r-l=q-p$ and swap $b_p$ and $b_q$.

Is it possible to make both arrays the same?

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 2 \cdot 10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$) — the length of the arrays $a$ and $b$.

The second line of each test case contains $n$ distinct integers $a_1,a_2,a_3,\ldots,a_n$ ($1 \le a_i \le 2 \cdot 10^5$) — the integers in the array $a$.

The third line of each test case contains $n$ distinct integers $b_1,b_2,b_3,\ldots,b_n$ ($1 \le b_i \le 2 \cdot 10^5$) — the integers in the array $b$.

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

For each testcase, print "YES" if the arrays $a$ and $b$ can be made the same. Otherwise, print "NO". can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

In the first testcase, you don't need to perform any operations since the arrays are same.

In the second testcase, it can be proven there exists no way to make the arrays same.

In the third testcase, one of the ways to make the arrays same is to first choose $l=1$, $r=3$, $p=1$, $q=3$ then choose $l=1$, $r=2$, $p=3$, $q=4$.
You are given a tree with $n$ nodes numbered from $1$ to $n$, along with an array of size $n$. The value of $i$-th node is $a_{i}$. There are $q$ queries. In each query, you are given 2 nodes numbered as $x$ and $y$.

Consider the path from the node numbered as $x$ to the node numbered as $y$. Let the path be represented by $x = p_0, p_1, p_2, \ldots, p_r = y$, where $p_i$ are the intermediate nodes. Compute the sum of $a_{p_i}\oplus i$ for each $i$ such that $0 \le i \le r$ where $\oplus$ is the [XOR](https://en.wikipedia.org/wiki/Exclusive_or) operator.

More formally, compute $$\sum_{i =0}^{r} a_{p_i}\oplus i$$.

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Each test case contains several sets of input data.

The first line of each set of input data contains a single integer $n$ ($1 \le n \le 5 \cdot 10^5$) — the number of nodes.

The next $n-1$ lines of each set of input data contain $2$ integers, $u$ and $v$ representing an edge between the node numbered $u$ and the node numbered $v$. It is guaranteed that $u \ne v$ and that the edges form a tree.

The next line of each set of input data contains $n$ integers, $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 5 \cdot 10^5$) — values of the nodes.

The next line contains a single integer $q$ ($1 \le q \le 10^5$) — the number of queries.

The next $q$ lines describe the queries. The $i$-th query contains $2$ integers $x$ and $y$ ($1 \le x,y \le n$) denoting the starting and the ending node of the path.

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

For each query, output a single number — the sum from the problem statement.

A digit is large if it is between $5$ and $9$, inclusive. A positive integer is large if all of its digits are large.

You are given an integer $x$. Can it be the sum of two large positive integers with the same number of digits?

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

For each test case, output a single integer — the sum of the weights of the edges of the minimum spanning tree of the graph $G$. If the graph $G$ has no spanning tree, output $-1$.

In the first test case, the graph $G$ is as follows:

![](CDN_BASE_URL/7b0a32f8f1d54a5dbbaffee1efd8ece1)

One of the minimum spanning trees of $G$ is as follows:

![](CDN_BASE_URL/7a035af2d5227648781d0c24b81c076e)

The sum of the weights of the edges of the minimum spanning tree is $9$.

In the second test case, the graph $G$ is as follows:

![](CDN_BASE_URL/62be880a08fa85071b491d6badd5f58e)

$G$ is already a tree, and the sum of the weights of the tree is $13$.

In the third test case, the graph $G$ is as follows:

![](CDN_BASE_URL/b667aa771d019e3a8a3a4c4372200f03)

In the fourth test case, the graph $G$ is as follows:

![](CDN_BASE_URL/45bc2d052677dbf032ff37fd1d723d83)

It's easy to see that $G$ is no

On another boring day, Egor got bored and decided to do something. But since he has no friends, he came up with a game to play.

Egor has a deck of $n$ cards, the $i$-th card from the top has a number $a_i$ written on it. Egor wants to play a certain number of rounds until the cards run out. In each round, he takes a non-zero number of cards from the top of the deck and finishes the round. If the sum of the numbers on the cards collected during the round is between $l$ and $r$, inclusive, the round is won; otherwise, it is lost.

Egor knows by heart the order of the cards. Help Egor determine the maximum number of rounds he can win in such a game. Note that Egor is not required to win rounds consecutively.

Each test consists of several test cases. The first line contains an integer $t$ ($1 \le t \le 10^{4}$) — the number of test cases. This is followed by a description of the test cases.

The first line of each test case contains three integers $n$, $l$, and $r$ ($1 \le n \le 10^{5}$, $1 \le l \le r \le 10^9$).

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 numbers on the cards from top to bottom.

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

For each test case, output a single number — the maximum number of rounds Egor can win.

In the first test case, Egor can win $3$ rounds:

* In the first round, take the top $2$ cards with values $2$ and $1$ and win, as their sum is $3$. After this, the deck will look like this: $[11, 3, 7]$. * In the second round, take the top card and lose, as its value $11$ is greater than $r = 10$. After this, the deck will look like this: $[3, 7]$. * In the third round, take the top card with value $3$ and win. After this, the deck will look like this: $[7]$. * After this, in the fourth round, Egor only has to take the last card in the deck with value $7$ and win again.

In the second test case, Egor cannot win any rounds, no matter how hard he tries.

In the t

Given two arrays of distinct positive integers $a$ and $b$ of length $n$, we would like to make both the arrays the same. Two arrays $x$ and $y$ of length $k$ are said to be the same when for all $1 \le i \le k$, $x_i = y_i$.

Now in one move, you can choose some index $l$ and $r$ in $a$ ($l \le r$) and swap $a_l$ and $a_r$, then choose some $p$ and $q$ ($p \le q$) in $b$ such that $r-l=q-p$ and swap $b_p$ and $b_q$.

Is it possible to make both arrays the same?

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 2 \cdot 10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$) — the length of the arrays $a$ and $b$.

The second line of each test case contains $n$ distinct integers $a_1,a_2,a_3,\ldots,a_n$ ($1 \le a_i \le 2 \cdot 10^5$) — the integers in the array $a$.

The third line of each test case contains $n$ distinct integers $b_1,b_2,b_3,\ldots,b_n$ ($1 \le b_i \le 2 \cdot 10^5$) — the integers in the array $b$.

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

For each testcase, print "YES" if the arrays $a$ and $b$ can be made the same. Otherwise, print "NO". can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

In the first testcase, you don't need to perform any operations since the arrays are same.

In the second testcase, it can be proven there exists no way to make the arrays same.

In the third testcase, one of the ways to make the arrays same is to first choose $l=1$, $r=3$, $p=1$, $q=3$ then choose $l=1$, $r=2$, $p=3$, $q=4$.

You are given a tree with $n$ nodes numbered from $1$ to $n$, along with an array of size $n$. The value of $i$-th node is $a_{i}$. There are $q$ queries. In each query, you are given 2 nodes numbered as $x$ and $y$.

Consider the path from the node numbered as $x$ to the node numbered as $y$. Let the path be represented by $x = p_0, p_1, p_2, \ldots, p_r = y$, where $p_i$ are the intermediate nodes. Compute the sum of $a_{p_i}\oplus i$ for each $i$ such that $0 \le i \le r$ where $\oplus$ is the [XOR](https://en.wikipedia.org/wiki/Exclusive_or) operator.

More formally, compute $$\sum_{i =0}^{r} a_{p_i}\oplus i$$.

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Each test case contains several sets of input data.

The first line of each set of input data contains a single integer $n$ ($1 \le n \le 5 \cdot 10^5$) — the number of nodes.

The next $n-1$ lines of each set of input data contain $2$ integers, $u$ and $v$ representing an edge between the node numbered $u$ and the node numbered $v$. It is guaranteed that $u \ne v$ and that the edges form a tree.

The next line of each set of input data contains $n$ integers, $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 5 \cdot 10^5$) — values of the nodes.

The next line contains a single integer $q$ ($1 \le q \le 10^5$) — the number of queries.

The next $q$ lines describe the queries. The $i$-th query contains $2$ integers $x$ and $y$ ($1 \le x,y \le n$) denoting the starting and the ending node of the path.

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

For each query, output a single number — the sum from the problem statement.

A digit is large if it is between $5$ and $9$, inclusive. A positive integer is large if all of its digits are large.

You are given an integer $x$. Can it be the sum of two large positive integers with the same number of digits?

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