CF1630D.Flipping Range

You are given an array $a$ of $n$ integers and a set $B$ of $m$ positive integers such that $1 \leq b_i \leq \lfloor \frac{n}{2} \rfloor$ for $1\le i\le m$ , where $b_i$ is the $i$ -th element of $B$ .

You can make the following operation on $a$ :

1. Select some $x$ such that $x$ appears in $B$ .
2. Select an interval from array $a$ of size $x$ and multiply by $-1$ every element in the interval. Formally, select $l$ and $r$ such that $1\leq l\leq r \leq n$ and $r-l+1=x$ , then assign $a_i:=-a_i$ for every $i$ such that $l\leq i\leq r$ .

Consider the following example, let $a=[0,6,-2,1,-4,5]$ and $B=\{1,2\}$ :

1. $[0,6,-2,-1,4,5]$ is obtained after choosing size $2$ and $l=4$ , $r=5$ .
2. $[0,6,2,-1,4,5]$ is obtained after choosing size $1$ and $l=3$ , $r=3$ .

Find the maximum $\sum\limits_{i=1}^n {a_i}$ you can get after applying such operation any number of times (possibly zero).

The input consists of multiple test cases. The first line contains a single integer $t$ ( $1 \le t \le 10^5$ ) — the number of test cases. Description of the test cases follows.

The first line of each test case contains two integers $n$ and $m$ ( $2\leq n \leq 10^6$ , $1 \leq m \leq \lfloor \frac{n}{2} \rfloor$ ) — the number of elements of $a$ and $B$ respectively.

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ( $-10^9\leq a_i \leq 10^9$ ).

The third line of each test case contains $m$ distinct positive integers $b_1,b_2,\ldots,b_m$ ( $1 \leq b_i \leq \lfloor \frac{n}{2} \rfloor$ ) — the elements in the set $B$ .

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

For each test case print a single integer — the maximum possible sum of all $a_i$ after applying such operation any number of times.

In the first test, you can apply the operation $x=1$ , $l=3$ , $r=3$ , and the operation $x=1$ , $l=5$ , $r=5$ , then the array becomes $[0, 6, 2, 1, 4, 5]$ .

In the second test, you can apply the operation $x=2$ , $l=2$ , $r=3$ , and the array becomes $[1, 1, -1, -1, 1, -1, 1]$ , then apply the operation $x=2$ , $l=3$ , $r=4$ , and the array becomes $[1, 1, 1, 1, 1, -1, 1]$ . There is no way to achieve a sum bigger than $5$ .