You are given a tree with $$$n$$$ vertices, rooted at vertex $$$1$$$. The $$$i$$$-th vertex has an integer $$$a_i$$$ written on it.
Let $$$L$$$ be the set of all direct children$$$^{\text{∗}}$$$ of $$$v$$$. A tree is called wonderful, if for all vertices $$$v$$$ where $$$L$$$ is not empty, $$$$$$a_v \le \sum_{u \in L}{a_u}.$$$$$$ In one operation, you choose any vertex $$$v$$$ and increase $$$a_v$$$ by $$$1$$$.
Find the minimum number of operations needed to make the given tree wonderful!
$$$^{\text{∗}}$$$ Vertex $$$u$$$ is called a direct child of vertex $$$v$$$ if:
- $$$u$$$ and $$$v$$$ are connected by an edge, and
- $$$v$$$ is on the (unique) path from $$$u$$$ to the root of the tree.
Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 5000$$$) — the number of vertices in the tree.
The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le 10^9$$$) — the values initially written on the vertices.
The third line of each test case contains $$$n - 1$$$ integers $$$p_2, p_3 , \ldots, p_n$$$ ($$$1 \le p_i < i$$$), indicating that there is an edge from vertex $$$p_i$$$ to vertex $$$i$$$. It is guaranteed that the given edges form a tree.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$5000$$$.
For each test case, output a single integer — the minimum number of operations needed to make the tree wonderful.
459 3 4 1 21 1 3 325 31236 54130 0 01 2
3 2 0 0
The tree in the first test case:
You can apply the operation once on vertex $$$5$$$ and twice on vertex $$$2$$$ to get a wonderful tree.
In the second test case, you can apply the operation twice on vertex $$$2$$$ to get a wonderful tree.
In the third and fourth test cases, the tree is already wonderful, so you don't need to apply any operations.
