You are given a rooted tree consisting of $n$ vertices. The vertices are numbered from $1$ to $n$, and the root is the vertex $1$. You are also given a score array $s_1, s_2, \ldots, s_n$.

A multiset of $k$ simple paths is called valid if the following two conditions are both true.

Each path starts from $1$.

Let $c_i$ be the number of paths covering vertex $i$. For each pair of vertices $(u,v)$ ($2\le u,v\le n$) that have the same parent, $|c_u-c_v|\le 1$ holds.

The value of the path multiset is defined as $\sum\limits_{i=1}^n c_i s_i$.

It can be shown that it is always possible to find at least one valid multiset. Find the maximum value among all valid multisets.


-----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.


-----Examples-----

Input
2
5 4
1 2 1 3
6 2 1 5 7
5 3
1 2 1 3
6 6 1 4 10
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 $3\cdot 6+ 2\cdot 6+2\cdot 1+1\cdot 4+2\cdot 10=56$.

Difficulty: HARD
Tags: Graph traversal

Sorting

Greedy algorithms

Tree algorithms

Dynamic programming
Skill types: Dynamic programming

Sorting

Greedy algorithms

Starter code:

Solutions:
from sys import stdin, setrecursionlimit
input = stdin.readline
inp = lambda : list(map(int, input().split()))
setrecursionlimit(3 * 10 ** 5)
import threading

def dfs(p, k):
	if (p, k) in dp:
		return dp[p, k]
	(count, take) = (len(child[p]), [])
	for i in child[p]:
		if k % count:
			v1 = dfs(i, k // count + 1)
			v2 = dfs(i, k // count)
			take.append([v1 - v2, v1, v2])
		else:
			v = dfs(i, k // count)
			take.append([0, 0, v])
	value = k * s[p - 1]
	if len(take):
		take.sort(reverse=True)
		(d, extra) = (k // count, k % count)
		for i in take:
			if extra:
				value += i[1]
				extra -= 1
			else:
				value += i[2]
	dp[p, k] = value
	return value

def answer():
	ans = dfs(1, k)
	return ans

def main():
	global child, n, k, p, s, dp
	for T in range(int(input())):
		(n, k) = inp()
		p = inp()
		child = [[] for i in range(n + 1)]
		for i in range(n - 1):
			child[p[i]].append(i + 2)
		s = inp()
		dp = dict()
		print(answer())
threading.stack_size(10 ** 8)
threading.Thread(target=main).start()