knightnemo/LMCompressionDataset · Datasets at Hugging Face

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![](CDN_BASE_URL/ff5ae9758c965c2d8398c936e9581dab) An example of the robot's movement process, $s = \texttt{"ULULURD"}$

The script $s$ will be executed for $k$ times continuously. All changes to the string $s$ will be retained even when it is repeated. During this process, how many times will the robot move to the point $(0, 0)$ in total? Note that the initial position does NOT count.

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

The first line of each test case contains four integers $n$, $k$, $w$, and $h$ ($1 \le n, w, h \le 10^6$; $1 \le k \le 10^{12}$).

The second line contains a single string $s$ of size $n$ ($s_i \in \\{\texttt{L}, \texttt{R}, \texttt{U}, \texttt{D}\\}$) — the script to be executed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$.

For each test case, print a single integer — the number of times the robot reaches $(0, 0)$ when executing script $s$ for $k$ times continuously.

In the first test case, the robot only moves up and right for the first two executions. After th
Yaroslav is playing a computer game, and at one of the levels, he encountered $n$ mushrooms arranged in a row. Each mushroom has its own level of toxicity; the $i$-th mushroom from the beginning has a toxicity level of $a_i$. Yaroslav can choose two integers $1 \le l \le r \le n$, and then his character will take turns from left to right to eat mushrooms from this subsegment one by one, i.e., the mushrooms with numbers $l, l+1, l+2, \ldots, r$.

The character has a toxicity level $g$, initially equal to $0$. The computer game is defined by the number $x$ — the maximum toxicity level at any given time. When eating a mushroom with toxicity level $k$, the following happens:

1. The toxicity level of the character is increased by $k$. 2. If $g \leq x$, the process continues; otherwise, $g$ becomes zero and the process continues.

Yaroslav became interested in how many ways there are to choose the values of $l$ and $r$ such that the final value of $g$ is not zero. Help Yaroslav find this number!

Each test consists of multiple test cases. The first line contains an integer $t$ ($1 \le t \le 10^{4}$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $n$, $x$ ($1 \leq n \leq 2 \cdot 10^5, 1 \le x \le 10^9$) — the number of mushrooms and the maximum toxicity level.

The second line of each test case contains $n$ numbers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$).

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

For each test case, output a single number — the number of subsegments such that the final value of $g$ will not be zero.

In the first test case, the subsegments $(1, 1)$, $(1, 2)$, $(1, 4)$, $(2, 2)$, $(2, 3)$, $(3, 3)$, $(3, 4)$ and $(4, 4)$ are suitable.

In the second test case, non-zero $g$ will remain only on the subsegments $(1, 1)$ and $(2, 2)$.

In the third test case, on the only possible subsegment, $g$ will be zero.
After winning another Bed Wars game, Masha and Olya wanted to relax and decided to play a new game. Masha gives Olya an array $a$ of length $n$ and a number $s$. Now Olya's task is to find a non-negative number $x$ such that $\displaystyle\sum_{i=1}^{n} a_i \oplus x = s$. But she is very tired after a tight round, so please help her with this.

But this task seemed too simple to them, so they decided to make the numbers larger (up to $2^k$) and provide you with their binary representation.

Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le n, k, n \cdot k \le 2 \cdot 10^6$) — the length of the array $a$ and the length of the binary representation of all numbers.

The second line contains a string of length $k$, consisting of zeros and ones — the binary representation of the number $s$, starting from the most significant bits.

The next $n$ lines also contain strings of length $k$, consisting of zeros and ones, the $i$-th of these strings contains the binary representation of the number $a_i$, starting from the most significant bits.

It is guaranteed that the sum of the values $n \cdot k$ for all test cases does not exceed $2 \cdot 10^6$.

For each test case, output a string of length $k$ on a separate line, consisting of zeros or ones — the binary representation of any suitable number $x$ ($x \ge 0$), starting from the most significant bits, or $-1$ if such $x$ does not exist.

In the first test case, $s = 11, a = [14, 6, 12, 15]$, if $x = 14$, then $\displaystyle\sum_{i=1}^{n} a_i \oplus x = (14 \oplus 14) + (6 \oplus 14) + (12 \oplus 14) + (15 \oplus 14) = 0 + 8 + 2 + 1 = 11 = s$.

In the second test case, $s = 41, a = [191, 158]$, if $x = 154$, then $\displaystyle\sum_{i=1}^{n} a_i \oplus x = (191 \oplus 154) + (158 \oplus 154) = 37 + 4 = 41 = s$.
This is an interactive problem!

Timofey is writing a competition called Capture the Flag (or CTF for short). He has one task left, which involves hacking a security system. The entire system is based on polynomial hashes$^{\text{∗}}$.

Timofey can input a string consisting of lowercase Latin letters into the system, and the system will return its polynomial hash. To hack the system, Timofey needs to find the polynomial hash parameters ($p$ and $m$) that the system uses.

Timofey doesn't have much time left, so he will only be able to make $3$ queries. Help him solve the task.

$^{\text{∗}}$ The polynomial hash of a string $s$, consisting of lowercase Latin letters of length $n$, based on $p$ and modulo $m$ is $(\mathrm{ord}(s_1) \cdot p ^ 0 + \mathrm{ord}(s_2) \cdot p ^ 1 + \mathrm{ord}(s_3) \cdot p ^ 2 + \ldots + \mathrm{ord}(s_n) \cdot p ^ {n - 1}) \bmod m$. Where $s_i$ denotes the $i$-th character of the string $s$, $\mathrm{ord}(\mathrm{chr})$ denotes the ordinal number of the character $\mathrm{chr}$ in the English alphabet, and $x \bmod m$ is the remainder of $x$ when divided by $m$.

Each test consists of multiple test cases. The first line contains an integer $t$ ($1 \leq t \leq 10^3$) — the number of test cases.

It is guaranteed that the $p$ and $m$ used by the system satisfy the conditions: $26 < p \leq 50$ and $p + 1 < m \leq 2 \cdot 10^9$.



Answer for the first query is $(ord(a) \cdot 31^0 + ord(a) \cdot 31^1) \mod 59 = (1 + 1 \cdot 31) \mod 59 = 32$.

Answer for the second query is $(ord(y) \cdot 31^0 + ord(b) \cdot 31^1) \mod 59 = (25 + 2 \cdot 31) \mod 59 = 28$.
Fox loves permutations! She came up with the following problem and asked Cat to solve it:

You are given an even positive integer $n$ and a permutation$^\dagger$ $p$ of length $n$.

The score of another permutation $q$ of length $n$ is the number of local maximums in the array $a$ of length $n$, where $a_i = p_i + q_i$ for all $i$ ($1 \le i \le n$). In other words, the score of $q$ is the number of $i$ such that $1 < i < n$ (note the strict inequalities), $a_{i-1} < a_i$, and $a_i > a_{i+1}$ (once again, note the strict inequalities).

Find the permutation $q$ that achieves the maximum score for given $n$ and $p$. If there exist multiple such permutations, you can pick any of them.

$^\dagger$ A permutation of length $n$ 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).

The first line of input contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases in the input you will have to solve.

The first line of each test case contains one even integer $n$ ($4 \leq n \leq 10^5$, $n$ is even) — the length of the permutation $p$.

The second line of each test case contains the $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$). It is guaranteed that $p$ is a permutation of length $n$.

It is guaranteed that the sum of $n$ across all test cases doesn't exceed $10^5$.

For each test case, output one line containing any permutation of length $n$ (the array $q$), such that $q$ maximizes the score under the given constraints.

In the first example, $a = [3, 6, 4, 7]$. The array has just one local maximum (on the second position), so the score of the chosen permutation $q$ is $1$. It can be proven that this score is optimal under the constraints.

In the last example, the resulting array $a = [6, 6, 12, 7, 14, 7, 14, 6]$ has $3$ local maximums — on the third, fifth and seven
Sparkle gives you two arrays $a$ and $b$ of length $n$. Initially, your score is $0$. In one operation, you can choose an integer $i$ and add $a_i$ to your score. Then, you must set $a_i$ = $\max(0, a_i - b_i)$.

![](CDN_BASE_URL/ff5ae9758c965c2d8398c936e9581dab) An example of the robot's movement process, $s = \texttt{"ULULURD"}$

The script $s$ will be executed for $k$ times continuously. All changes to the string $s$ will be retained even when it is repeated. During this process, how many times will the robot move to the point $(0, 0)$ in total? Note that the initial position does NOT count.

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

The first line of each test case contains four integers $n$, $k$, $w$, and $h$ ($1 \le n, w, h \le 10^6$; $1 \le k \le 10^{12}$).

The second line contains a single string $s$ of size $n$ ($s_i \in \\{\texttt{L}, \texttt{R}, \texttt{U}, \texttt{D}\\}$) — the script to be executed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$.

For each test case, print a single integer — the number of times the robot reaches $(0, 0)$ when executing script $s$ for $k$ times continuously.

In the first test case, the robot only moves up and right for the first two executions. After th

Yaroslav is playing a computer game, and at one of the levels, he encountered $n$ mushrooms arranged in a row. Each mushroom has its own level of toxicity; the $i$-th mushroom from the beginning has a toxicity level of $a_i$. Yaroslav can choose two integers $1 \le l \le r \le n$, and then his character will take turns from left to right to eat mushrooms from this subsegment one by one, i.e., the mushrooms with numbers $l, l+1, l+2, \ldots, r$.

The character has a toxicity level $g$, initially equal to $0$. The computer game is defined by the number $x$ — the maximum toxicity level at any given time. When eating a mushroom with toxicity level $k$, the following happens:

1. The toxicity level of the character is increased by $k$. 2. If $g \leq x$, the process continues; otherwise, $g$ becomes zero and the process continues.

Yaroslav became interested in how many ways there are to choose the values of $l$ and $r$ such that the final value of $g$ is not zero. Help Yaroslav find this number!

Each test consists of multiple test cases. The first line contains an integer $t$ ($1 \le t \le 10^{4}$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $n$, $x$ ($1 \leq n \leq 2 \cdot 10^5, 1 \le x \le 10^9$) — the number of mushrooms and the maximum toxicity level.

The second line of each test case contains $n$ numbers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$).

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

For each test case, output a single number — the number of subsegments such that the final value of $g$ will not be zero.

In the first test case, the subsegments $(1, 1)$, $(1, 2)$, $(1, 4)$, $(2, 2)$, $(2, 3)$, $(3, 3)$, $(3, 4)$ and $(4, 4)$ are suitable.

In the second test case, non-zero $g$ will remain only on the subsegments $(1, 1)$ and $(2, 2)$.

In the third test case, on the only possible subsegment, $g$ will be zero.

After winning another Bed Wars game, Masha and Olya wanted to relax and decided to play a new game. Masha gives Olya an array $a$ of length $n$ and a number $s$. Now Olya's task is to find a non-negative number $x$ such that $\displaystyle\sum_{i=1}^{n} a_i \oplus x = s$. But she is very tired after a tight round, so please help her with this.

But this task seemed too simple to them, so they decided to make the numbers larger (up to $2^k$) and provide you with their binary representation.

Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le n, k, n \cdot k \le 2 \cdot 10^6$) — the length of the array $a$ and the length of the binary representation of all numbers.

The second line contains a string of length $k$, consisting of zeros and ones — the binary representation of the number $s$, starting from the most significant bits.

The next $n$ lines also contain strings of length $k$, consisting of zeros and ones, the $i$-th of these strings contains the binary representation of the number $a_i$, starting from the most significant bits.

It is guaranteed that the sum of the values $n \cdot k$ for all test cases does not exceed $2 \cdot 10^6$.

For each test case, output a string of length $k$ on a separate line, consisting of zeros or ones — the binary representation of any suitable number $x$ ($x \ge 0$), starting from the most significant bits, or $-1$ if such $x$ does not exist.

In the first test case, $s = 11, a = [14, 6, 12, 15]$, if $x = 14$, then $\displaystyle\sum_{i=1}^{n} a_i \oplus x = (14 \oplus 14) + (6 \oplus 14) + (12 \oplus 14) + (15 \oplus 14) = 0 + 8 + 2 + 1 = 11 = s$.

In the second test case, $s = 41, a = [191, 158]$, if $x = 154$, then $\displaystyle\sum_{i=1}^{n} a_i \oplus x = (191 \oplus 154) + (158 \oplus 154) = 37 + 4 = 41 = s$.

This is an interactive problem!

Timofey is writing a competition called Capture the Flag (or CTF for short). He has one task left, which involves hacking a security system. The entire system is based on polynomial hashes$^{\text{∗}}$.

Timofey can input a string consisting of lowercase Latin letters into the system, and the system will return its polynomial hash. To hack the system, Timofey needs to find the polynomial hash parameters ($p$ and $m$) that the system uses.

Timofey doesn't have much time left, so he will only be able to make $3$ queries. Help him solve the task.

$^{\text{∗}}$ The polynomial hash of a string $s$, consisting of lowercase Latin letters of length $n$, based on $p$ and modulo $m$ is $(\mathrm{ord}(s_1) \cdot p ^ 0 + \mathrm{ord}(s_2) \cdot p ^ 1 + \mathrm{ord}(s_3) \cdot p ^ 2 + \ldots + \mathrm{ord}(s_n) \cdot p ^ {n - 1}) \bmod m$. Where $s_i$ denotes the $i$-th character of the string $s$, $\mathrm{ord}(\mathrm{chr})$ denotes the ordinal number of the character $\mathrm{chr}$ in the English alphabet, and $x \bmod m$ is the remainder of $x$ when divided by $m$.

Each test consists of multiple test cases. The first line contains an integer $t$ ($1 \leq t \leq 10^3$) — the number of test cases.

It is guaranteed that the $p$ and $m$ used by the system satisfy the conditions: $26 < p \leq 50$ and $p + 1 < m \leq 2 \cdot 10^9$.

Answer for the first query is $(ord(a) \cdot 31^0 + ord(a) \cdot 31^1) \mod 59 = (1 + 1 \cdot 31) \mod 59 = 32$.

Answer for the second query is $(ord(y) \cdot 31^0 + ord(b) \cdot 31^1) \mod 59 = (25 + 2 \cdot 31) \mod 59 = 28$.

Fox loves permutations! She came up with the following problem and asked Cat to solve it:

You are given an even positive integer $n$ and a permutation$^\dagger$ $p$ of length $n$.

The score of another permutation $q$ of length $n$ is the number of local maximums in the array $a$ of length $n$, where $a_i = p_i + q_i$ for all $i$ ($1 \le i \le n$). In other words, the score of $q$ is the number of $i$ such that $1 < i < n$ (note the strict inequalities), $a_{i-1} < a_i$, and $a_i > a_{i+1}$ (once again, note the strict inequalities).

Find the permutation $q$ that achieves the maximum score for given $n$ and $p$. If there exist multiple such permutations, you can pick any of them.

$^\dagger$ A permutation of length $n$ 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).

The first line of input contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases in the input you will have to solve.

The first line of each test case contains one even integer $n$ ($4 \leq n \leq 10^5$, $n$ is even) — the length of the permutation $p$.

The second line of each test case contains the $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$). It is guaranteed that $p$ is a permutation of length $n$.

It is guaranteed that the sum of $n$ across all test cases doesn't exceed $10^5$.

For each test case, output one line containing any permutation of length $n$ (the array $q$), such that $q$ maximizes the score under the given constraints.

In the first example, $a = [3, 6, 4, 7]$. The array has just one local maximum (on the second position), so the score of the chosen permutation $q$ is $1$. It can be proven that this score is optimal under the constraints.

In the last example, the resulting array $a = [6, 6, 12, 7, 14, 7, 14, 6]$ has $3$ local maximums — on the third, fifth and seven

Sparkle gives you two arrays $a$ and $b$ of length $n$. Initially, your score is $0$. In one operation, you can choose an integer $i$ and add $a_i$ to your score. Then, you must set $a_i$ = $\max(0, a_i - b_i)$.