Maximum Monogonosity

2000ms

524288K

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 $$$k$$$.

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.

Sample Input:

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

Sample Output:

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 $$$[4, 4]$$$.

Informação

Provedor Codeforces

Código CF1859E