knightnemo/LMCompressionDataset · Datasets at Hugging Face

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Each test contains multiple test cases. The first line of input contains a single 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 a single integer $n$ ($1 \le n \le 10^5$) — the length of the array $a$.

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 array $a$.

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

For each test case, output a single integer — the minimum number of coins needed to make $a$ non-decreasing.

In the first test case, $a$ is already sorted, so you don't have to spend any coins.

In the second test case, the optimal sequence of operations is:

* Choose $k = 2$ and the indices $2$ and $5$: $[ 2, \color{red}{1}, 4, 7, \color{red}{6} ] \rightarrow [2, 2, 4, 7, 7]$. This costs $3$ coins.

It can be proven that it is not possible to make $a$ non-decreasing by spending less than $3$ coins.
This is the easy version of the problem. The only difference between the two versions is the constraint on $n$. You can make hacks only if both versions of the problem are solved.

You are given an array of integers $a$ of length $n$.

In one operation, you will perform the following two-step process:

1. Choose an index $i$ such that $1 \le i < \|a\|$ and $a_i = i$. 2. Remove $a_i$ and $a_{i+1}$ from the array and concatenate the remaining parts.

Find the maximum number of times that you can perform the operation above.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 100$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 100$) — the length of the array $a$.

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

It is guaranteed that the sum of $n$ over all test cases does not exceed $100$.

For each test case, output a single integer — the maximum number of times that you can perform the operation.

In the first test case, one possible optimal sequence of operations is $[ 1, 5, \color{red}{3}, \color{red}{2}, 4 ] \rightarrow [\color{red}{1}, \color{red}{5}, 4] \rightarrow [4]$.

In the third test case, one possible optimal sequence of operations is $[1, \color{red}{2}, \color{red}{3}] \rightarrow [1]$.
This temple only magnifies the mountain's power.

Badeline

This is an interactive problem.

You are given two positive integers $n$ and $m$ ($\bf{n \le m}$).

The jury has hidden from you a rectangular matrix $a$ with $n$ rows and $m$ columns, where $a_{i,j} \in \\{ -1, 0, 1 \\}$ for all $1 \le i \le n$ and $1 \le j \le m$. The jury has also selected a cell $(i_0, j_0)$. Your goal is to find $(i_0,j_0)$.

In one query, you give a cell $(i, j)$, then the jury will reply with an integer.

* If $(i, j) = (i_0, j_0)$, the jury will reply with $0$. * Else, let $S$ be the sum of $a_{x,y}$ over all $x$ and $y$ such that $\min(i, i_0) \le x \le \max(i, i_0)$ and $\min(j, j_0) \le y \le \max(j, j_0)$. Then, the jury will reply with $\|i - i_0\| + \|j - j_0\| + \|S\|$.

Find $(i_0, j_0)$ by making at most $n + 225$ queries.

Note: the grader is not adaptive: $a$ and $(i_0,j_0)$ are fixed before any queries are made.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 50$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $n$ and $m$ ($1 \le n \le m \le 5000$) — the numbers of rows and the number of columns of the hidden matrix $a$ respectively.

It is guaranteed that the sum of $n \cdot m$ over all test cases does not exceed $25 \cdot 10^6$.



The hidden matrix in the first test case:

$1$\| $0$\| $1$\| $\color{red}{\textbf{0}}$ ---\|---\|---\|--- $1$\| $0$\| $0$\| $1$ $0$\| $-1$\| $-1$\| $-1$ The hidden matrix in the second test case:

$\color{red}{\textbf{0}}$ --- Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.
For an array $[a_1,a_2,\ldots,a_n]$ of length $n$, define $f(a)$ as the sum of the minimum element over all subsegments. That is, $$f(a)=\sum_{l=1}^n\sum_{r=l}^n \min_{l\le i\le r}a_i.$$

A permutation is a sequence of integers from $1$ to $n$ of length $n$ containing each number exactly once. You are given a permutation $[a_1,a_2,\ldots,a_n]$. For each $i$, solve the following problem independently:

* Erase $a_i$ from $a$, concatenating the remaining parts, resulting in $b = [a_1,a_2,\ldots,a_{i-1},\;a_{i+1},\ldots,a_{n}]$. * Calculate $f(b)$.

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

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

The second line of each test case contains $n$ distinct integers $a_1,\ldots,a_n$ ($1\le a_i\le n$).

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

For each test case, print one line containing $n$ integers. The $i$-th integer should be the answer when erasing $a_i$.

In the second test case, $a=[3,1,2]$.

* When removing $a_1$, $b=[1,2]$. $f(b)=1+2+\min\\{1,2\\}=4$. * When removing $a_2$, $b=[3,2]$. $f(b)=3+2+\min\\{3,2\\}=7$. * When removing $a_3$, $b=[3,1]$. $f(b)=3+1+\min\\{3,1\\}=5$.
For an array $u_1, u_2, \ldots, u_n$, define

* a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$; * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$; * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.

Each test contains multiple test cases. The first line of input contains a single 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 a single integer $n$ ($1 \le n \le 10^5$) — the length of the array $a$.

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 array $a$.

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

For each test case, output a single integer — the minimum number of coins needed to make $a$ non-decreasing.

In the first test case, $a$ is already sorted, so you don't have to spend any coins.

In the second test case, the optimal sequence of operations is:

* Choose $k = 2$ and the indices $2$ and $5$: $[ 2, \color{red}{1}, 4, 7, \color{red}{6} ] \rightarrow [2, 2, 4, 7, 7]$. This costs $3$ coins.

It can be proven that it is not possible to make $a$ non-decreasing by spending less than $3$ coins.

This is the easy version of the problem. The only difference between the two versions is the constraint on $n$. You can make hacks only if both versions of the problem are solved.

You are given an array of integers $a$ of length $n$.

In one operation, you will perform the following two-step process:

1. Choose an index $i$ such that $1 \le i < |a|$ and $a_i = i$. 2. Remove $a_i$ and $a_{i+1}$ from the array and concatenate the remaining parts.

Find the maximum number of times that you can perform the operation above.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 100$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 100$) — the length of the array $a$.

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

It is guaranteed that the sum of $n$ over all test cases does not exceed $100$.

For each test case, output a single integer — the maximum number of times that you can perform the operation.

In the first test case, one possible optimal sequence of operations is $[ 1, 5, \color{red}{3}, \color{red}{2}, 4 ] \rightarrow [\color{red}{1}, \color{red}{5}, 4] \rightarrow [4]$.

In the third test case, one possible optimal sequence of operations is $[1, \color{red}{2}, \color{red}{3}] \rightarrow [1]$.

This temple only magnifies the mountain's power.

Badeline

This is an interactive problem.

You are given two positive integers $n$ and $m$ ($\bf{n \le m}$).

The jury has hidden from you a rectangular matrix $a$ with $n$ rows and $m$ columns, where $a_{i,j} \in \\{ -1, 0, 1 \\}$ for all $1 \le i \le n$ and $1 \le j \le m$. The jury has also selected a cell $(i_0, j_0)$. Your goal is to find $(i_0,j_0)$.

In one query, you give a cell $(i, j)$, then the jury will reply with an integer.

* If $(i, j) = (i_0, j_0)$, the jury will reply with $0$. * Else, let $S$ be the sum of $a_{x,y}$ over all $x$ and $y$ such that $\min(i, i_0) \le x \le \max(i, i_0)$ and $\min(j, j_0) \le y \le \max(j, j_0)$. Then, the jury will reply with $|i - i_0| + |j - j_0| + |S|$.

Find $(i_0, j_0)$ by making at most $n + 225$ queries.

Note: the grader is not adaptive: $a$ and $(i_0,j_0)$ are fixed before any queries are made.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ($1 \le t \le 50$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $n$ and $m$ ($1 \le n \le m \le 5000$) — the numbers of rows and the number of columns of the hidden matrix $a$ respectively.

It is guaranteed that the sum of $n \cdot m$ over all test cases does not exceed $25 \cdot 10^6$.

The hidden matrix in the first test case:

$1$| $0$| $1$| $\color{red}{\textbf{0}}$ ---|---|---|--- $1$| $0$| $0$| $1$ $0$| $-1$| $-1$| $-1$ The hidden matrix in the second test case:

$\color{red}{\textbf{0}}$ --- Note that the line breaks in the example input and output are for the sake of clarity, and do not occur in the real interaction.

For an array $[a_1,a_2,\ldots,a_n]$ of length $n$, define $f(a)$ as the sum of the minimum element over all subsegments. That is, $$f(a)=\sum_{l=1}^n\sum_{r=l}^n \min_{l\le i\le r}a_i.$$

A permutation is a sequence of integers from $1$ to $n$ of length $n$ containing each number exactly once. You are given a permutation $[a_1,a_2,\ldots,a_n]$. For each $i$, solve the following problem independently:

* Erase $a_i$ from $a$, concatenating the remaining parts, resulting in $b = [a_1,a_2,\ldots,a_{i-1},\;a_{i+1},\ldots,a_{n}]$. * Calculate $f(b)$.

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

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

The second line of each test case contains $n$ distinct integers $a_1,\ldots,a_n$ ($1\le a_i\le n$).

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

For each test case, print one line containing $n$ integers. The $i$-th integer should be the answer when erasing $a_i$.

In the second test case, $a=[3,1,2]$.

* When removing $a_1$, $b=[1,2]$. $f(b)=1+2+\min\\{1,2\\}=4$. * When removing $a_2$, $b=[3,2]$. $f(b)=3+2+\min\\{3,2\\}=7$. * When removing $a_3$, $b=[3,1]$. $f(b)=3+1+\min\\{3,1\\}=5$.

For an array $u_1, u_2, \ldots, u_n$, define

* a prefix maximum as an index $i$ such that $u_i>u_j$ for all $j<i$; * a suffix maximum as an index $i$ such that $u_i>u_j$ for all $j>i$; * an ascent as an index $i$ ($i>1$) such that $u_i>u_{i-1}$.

You are given three cost arrays: $[a_1, a_2, \ldots, a_n]$, $[b_1, b_2, \ldots, b_n]$, and $[c_0, c_1, \ldots, c_{n-1}]$.

Define the cost of an array that has $x$ prefix maximums, $y$ suffix maximums, and $z$ ascents as $a_x\cdot b_y\cdot c_z$.

Let the sum of costs of all permutations of $1,2,\ldots,n$ be $f(n)$. Find $f(1)$, $f(2)$, ..., $f(n)$ modulo $998\,244\,353$.