Paths on the Tree

Description:

Input:

Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two space-separated integers $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$) and $$$k$$$ ($$$1 \le k \le 10^9$$$) — the size of the tree and the required number of paths.

The second line contains $$$n - 1$$$ space-separated integers $$$p_2,p_3,\ldots,p_n$$$ ($$$1\le p_i\le n$$$), where $$$p_i$$$ is the parent of the $$$i$$$-th vertex. It is guaranteed that this value describe a valid tree with root $$$1$$$.

The third line contains $$$n$$$ space-separated integers $$$s_1,s_2,\ldots,s_n$$$ ($$$0 \le s_i \le 10^4$$$) — the scores of the vertices.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10 ^ 5$$$.

Output:

For each test case, print a single integer — the maximum value of a path multiset.

Sample Input:

2
5 4
1 2 1 3
6 2 1 5 7
5 3
1 2 1 3
6 6 1 4 10

Sample Output:

54
56

Note:

In the first test case, one of optimal solutions is four paths $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 4$$$, $$$1 \to 4$$$, here $$$c=[4,2,2,2,2]$$$. The value equals to $$$4\cdot 6+ 2\cdot 2+2\cdot 1+2\cdot 5+2\cdot 7=54$$$.

In the second test case, one of optimal solution is three paths $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 2 \to 3 \to 5$$$, $$$1 \to 4$$$, here $$$c=[3,2,2,1,2]$$$. The value equals to $$$3\cdot 6+ 2\cdot 6+2\cdot 1+1\cdot 4+2\cdot 10=56$$$.