codeforces#P1859E. Maximum Monogonosity
Maximum Monogonosity
Description
You are given an array $a$ of length $n$ and an array $b$ of length $n$. The cost of a segment $[l, r]$, $1 \le l \le r \le n$, is defined as $|b_l - a_r| + |b_r - a_l|$.
Recall that two segments $[l_1, r_1]$, $1 \le l_1 \le r_1 \le n$, and $[l_2, r_2]$, $1 \le l_2 \le r_2 \le n$, are non-intersecting if one of the following conditions is satisfied: $r_1 < l_2$ or $r_2 < l_1$.
The length of a segment $[l, r]$, $1 \le l \le r \le n$, is defined as $r - l + 1$.
Find the maximum possible sum of costs of non-intersecting segments $[l_j, r_j]$, $1 \le l_j \le r_j \le n$, whose total length is equal to $k$.
Each test consists of multiple test cases. The first line contains a single integer $t$ $(1 \le t \le 1000)$ — the number of sets of input data. The description of the test cases follows.
The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 3000$) — the length of array $a$ and the total length of segments.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($-10^9 \le a_i \le 10^9$) — the elements of array $a$.
The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($-10^9 \le b_i \le 10^9$) — the elements of array $b$.
It is guaranteed that the sum of $n$ over all test case does not exceed $3000$.
For each test case, output a single number — the maximum possible sum of costs of such segments.
Input
Each test consists of multiple test cases. The first line contains a single integer $t$ $(1 \le t \le 1000)$ — the number of sets of input data. The description of the test cases follows.
The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 3000$) — the length of array $a$ and the total length of segments.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($-10^9 \le a_i \le 10^9$) — the elements of array $a$.
The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($-10^9 \le b_i \le 10^9$) — the elements of array $b$.
It is guaranteed that the sum of $n$ over all test case does not exceed $3000$.
Output
For each test case, output a single number — the maximum possible sum of costs of such segments.
5
4 4
1 1 1 1
1 1 1 1
3 2
1 3 2
5 2 3
5 1
1 2 3 4 5
1 2 3 4 5
7 2
1 3 7 6 4 7 2
1 5 3 2 7 4 5
4 2
17 3 5 8
16 2 5 9
0
10
0
16
28
Note
In the first test case, the cost of any segment is $0$, so the total cost is $0$.
In the second test case, we can take the segment $[1, 1]$ with a cost of $8$ and the segment $[3, 3]$ with a cost of $2$ to get a total sum of $10$. It can be shown that this is the optimal solution.
In the third test case, we are only interested in segments of length $1$, and the cost of any such segment is $0$.
In the fourth test case, it can be shown that the optimal sum is $16$. For example, we can take the segments $[3, 3]$ and $[4, 4]$.