Lowest Common Ancestor

Bobo has a rooted tree with $n$ nodes which are conveniently labeled with $1, 2, \dots, n$. Node $1$ is the root, and the $i$-th node has weight $w\_i$.

He would like to find out $f(2), f(3), \dots, f(n)$ where

$f(i) = \sum\_{j = 1}^{i - 1} w\_{\mathrm{LCA}(i, j)}\text{.}$

The input contains zero or more test cases, and is terminated by end-of-file. For each test case:

The first line contains an integer $n$ ($2 \leq n \leq 2 \cdot 10^5$).

The second line contains $n$ integers $w\_1, w\_2, \dots, w\_n$ ($1 \leq w\_i \leq 10^4$).

The third line contains $(n - 1)$ integers $p\_2, p\_3, \dots, p\_n$, where $p\_i$ denotes an edge from the $p\_i$-th node to the $i$-th node ($1 \leq p\_i \leq n$). The edges form a tree.

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

For each test case, output $(n - 1)$ integers: $f(2), f(3), \dots, f(n)$.