knightnemo/LMCompressionDataset · Datasets at Hugging Face

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Turtle wants you to calculate the maximum number of inversions of all interesting permutations of length $n$, or tell him if there is no interesting permutation.

An inversion of a permutation $p$ is a pair of integers $(i, j)$ ($1 \le i < j \le n$) such that $p_i > p_j$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($2 \le n \le 5 \cdot 10^3, 0 \le m \le \frac{n}{2}$) — the length of the permutation and the number of intervals.

The $i$-th of the following $m$ lines contains two integers $l_i, r_i$ ($1 \le l_i < r_i \le n$) — the $i$-th interval.

Additional constraint on the input in this version: $r_i < l_{i + 1}$ holds for each $i$ from $1$ to $m - 1$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^3$.

For each test case, if there is no interesting permutation, output a single integer $-1$.

Otherwise, output a single integer — the maximum number of inversions.

In the third test case, the interesting permutation with the maximum number of inversions is $[5, 2, 4, 3, 1]$.

In the fourth test case, the interesting permutation with the maximum number of inversions is $[4, 8, 7, 6, 3, 2, 1, 5]$. In this case, we can let $[k
This is a hard version of this problem. The differences between the versions are the constraint on $m$ and $r_i < l_{i + 1}$ holds for each $i$ from $1$ to $m - 1$ in the easy version. You can make hacks only if both versions of the problem are solved.

Turtle gives you $m$ intervals $[l_1, r_1], [l_2, r_2], \ldots, [l_m, r_m]$. He thinks that a permutation $p$ is interesting if there exists an integer $k_i$ for every interval ($l_i \le k_i < r_i$), and if he lets $a_i = \max\limits_{j = l_i}^{k_i} p_j, b_i = \min\limits_{j = k_i + 1}^{r_i} p_j$ for every integer $i$ from $1$ to $m$, the following condition holds:

$$\max\limits_{i = 1}^m a_i < \min\limits_{i = 1}^m b_i$$

Turtle wants you to calculate the maximum number of inversions of all interesting permutations of length $n$, or tell him if there is no interesting permutation.

An inversion of a permutation $p$ is a pair of integers $(i, j)$ ($1 \le i < j \le n$) such that $p_i > p_j$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($2 \le n \le 5 \cdot 10^3, 0 \le m \le 5 \cdot 10^3$) — the length of the permutation and the number of intervals.

The $i$-th of the following $m$ lines contains two integers $l_i, r_i$ ($1 \le l_i < r_i \le n$) — the $i$-th interval. Note that there may exist identical intervals (i.e., there may exist two different indices $i, j$ such that $l_i = l_j$ and $r_i = r_j$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^3$ and the sum of $m$ over all test cases does not exceed $5 \cdot 10^3$.

For each test case, if there is no interesting permutation, output a single integer $-1$.

Otherwise, output a single integer — the maximum number of inversions.

In the third test case, the interesting permutation with the maximum number of inversions is $[5, 2, 4, 3, 1]$.

In the fourth test case, the interesting permutation
Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

* $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$; * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

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

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.
Consider a set of points on a line. The distance between two points $i$ and $j$ is $\|i - j\|$.

The point $i$ from the set is the closest to the point $j$ from the set, if there is no other point $k$ in the set such that the distance from $j$ to $k$ is strictly less than the distance from $j$ to $i$. In other words, all other points from the set have distance to $j$ greater or equal to $\|i - j\|$.

For example, consider a set of points $\\{1, 3, 5, 8\\}$:

* for the point $1$, the closest point is $3$ (other points have distance greater than $\|1-3\| = 2$); * for the point $3$, there are two closest points: $1$ and $5$; * for the point $5$, the closest point is $3$ (but not $8$, since its distance is greater than $\|3-5\|$); * for the point $8$, the closest point is $5$.

You are given a set of points. You have to add an integer point into this set in such a way that it is different from every existing point in the set, and it becomes the closest point to every point in the set. Is it possible?

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

Each test case consists of two lines:

* the first line contains one integer $n$ ($2 \le n \le 40$) — the number of points in the set; * the second line contains $n$ integers $x_1, x_2, \dots, x_n$ ($1 \le x_1 < x_2 < \dots < x_n \le 100$) — the points from the set.

For each test case, print YES if it is possible to add a new point according to the conditions from the statement. Otherwise, print NO.

In the first example, the point $7$ will be the closest to both $3$ and $8$.

In the second example, it is impossible to add an integer point so that it becomes the closest to both $5$ and $6$, and is different from both of them.
There are $100$ rooms arranged in a row and $99$ doors between them; the $i$-th door connects rooms $i$ and $i+1$. Each door can be either locked or unlocked. Initially, all doors are unlocked.

We say that room $x$ is reachable from room $y$ if all doors between them are unlocked.

You know that:

* Alice is in some room from the segment $[l, r]$; * Bob is in some room from the segment $[L, R]$; * Alice and Bob are in different rooms.

Turtle wants you to calculate the maximum number of inversions of all interesting permutations of length $n$, or tell him if there is no interesting permutation.

An inversion of a permutation $p$ is a pair of integers $(i, j)$ ($1 \le i < j \le n$) such that $p_i > p_j$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($2 \le n \le 5 \cdot 10^3, 0 \le m \le \frac{n}{2}$) — the length of the permutation and the number of intervals.

The $i$-th of the following $m$ lines contains two integers $l_i, r_i$ ($1 \le l_i < r_i \le n$) — the $i$-th interval.

Additional constraint on the input in this version: $r_i < l_{i + 1}$ holds for each $i$ from $1$ to $m - 1$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^3$.

For each test case, if there is no interesting permutation, output a single integer $-1$.

Otherwise, output a single integer — the maximum number of inversions.

In the third test case, the interesting permutation with the maximum number of inversions is $[5, 2, 4, 3, 1]$.

In the fourth test case, the interesting permutation with the maximum number of inversions is $[4, 8, 7, 6, 3, 2, 1, 5]$. In this case, we can let $[k

This is a hard version of this problem. The differences between the versions are the constraint on $m$ and $r_i < l_{i + 1}$ holds for each $i$ from $1$ to $m - 1$ in the easy version. You can make hacks only if both versions of the problem are solved.

Turtle gives you $m$ intervals $[l_1, r_1], [l_2, r_2], \ldots, [l_m, r_m]$. He thinks that a permutation $p$ is interesting if there exists an integer $k_i$ for every interval ($l_i \le k_i < r_i$), and if he lets $a_i = \max\limits_{j = l_i}^{k_i} p_j, b_i = \min\limits_{j = k_i + 1}^{r_i} p_j$ for every integer $i$ from $1$ to $m$, the following condition holds:

$$\max\limits_{i = 1}^m a_i < \min\limits_{i = 1}^m b_i$$

Turtle wants you to calculate the maximum number of inversions of all interesting permutations of length $n$, or tell him if there is no interesting permutation.

An inversion of a permutation $p$ is a pair of integers $(i, j)$ ($1 \le i < j \le n$) such that $p_i > p_j$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($2 \le n \le 5 \cdot 10^3, 0 \le m \le 5 \cdot 10^3$) — the length of the permutation and the number of intervals.

The $i$-th of the following $m$ lines contains two integers $l_i, r_i$ ($1 \le l_i < r_i \le n$) — the $i$-th interval. Note that there may exist identical intervals (i.e., there may exist two different indices $i, j$ such that $l_i = l_j$ and $r_i = r_j$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^3$ and the sum of $m$ over all test cases does not exceed $5 \cdot 10^3$.

For each test case, if there is no interesting permutation, output a single integer $-1$.

Otherwise, output a single integer — the maximum number of inversions.

In the third test case, the interesting permutation with the maximum number of inversions is $[5, 2, 4, 3, 1]$.

In the fourth test case, the interesting permutation

Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

* $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$; * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

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

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Consider a set of points on a line. The distance between two points $i$ and $j$ is $|i - j|$.

The point $i$ from the set is the closest to the point $j$ from the set, if there is no other point $k$ in the set such that the distance from $j$ to $k$ is strictly less than the distance from $j$ to $i$. In other words, all other points from the set have distance to $j$ greater or equal to $|i - j|$.

For example, consider a set of points $\\{1, 3, 5, 8\\}$:

* for the point $1$, the closest point is $3$ (other points have distance greater than $|1-3| = 2$); * for the point $3$, there are two closest points: $1$ and $5$; * for the point $5$, the closest point is $3$ (but not $8$, since its distance is greater than $|3-5|$); * for the point $8$, the closest point is $5$.

You are given a set of points. You have to add an integer point into this set in such a way that it is different from every existing point in the set, and it becomes the closest point to every point in the set. Is it possible?

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

Each test case consists of two lines:

* the first line contains one integer $n$ ($2 \le n \le 40$) — the number of points in the set; * the second line contains $n$ integers $x_1, x_2, \dots, x_n$ ($1 \le x_1 < x_2 < \dots < x_n \le 100$) — the points from the set.

For each test case, print YES if it is possible to add a new point according to the conditions from the statement. Otherwise, print NO.

In the first example, the point $7$ will be the closest to both $3$ and $8$.

In the second example, it is impossible to add an integer point so that it becomes the closest to both $5$ and $6$, and is different from both of them.

There are $100$ rooms arranged in a row and $99$ doors between them; the $i$-th door connects rooms $i$ and $i+1$. Each door can be either locked or unlocked. Initially, all doors are unlocked.

We say that room $x$ is reachable from room $y$ if all doors between them are unlocked.

You know that:

* Alice is in some room from the segment $[l, r]$; * Bob is in some room from the segment $[L, R]$; * Alice and Bob are in different rooms.