CF1859E.Maximum Monogonosity

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.

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 $[4, 4]$ .