Triameter
Description
A tree is a connected undirected graph without cycles. A weighted tree has a weight assigned to each edge. The degree of a vertex is the number of edges connected to this vertex.
You are given a weighted tree with $n$ vertices, each edge has a weight of $1$. Let $L$ be the set of vertices with degree equal to $1$.
You have to answer $q$ independent queries. In the $i$-th query:
- You are given a positive integer $x_i$.
- For all $u,v \in L$ such that $u < v$, add edge $(u, v)$ with weight $x_i$ to the graph (initially the given tree).
- Find the diameter of the resulting graph.
The diameter of a graph is equal to $\max\limits_{1 \le u < v \le n}{\operatorname{d}(u, v)}$, where $\operatorname{d}(u, v)$ is the length of the shortest path between vertex $u$ and vertex $v$.
The first line contains a single integer $n$ ($3 \le n \le 10^6$).
The second line contains $n - 1$ integers $p_2,p_3,\ldots,p_n$ ($1 \le p_i < i$) indicating that there is an edge between vertices $i$ and $p_i$. It is guaranteed that the given edges form a tree.
The third line contains a single integer $q$ ($1 \le q \le 10$).
The fourth line contains $q$ integers $x_1,x_2,\ldots,x_q$ ($1 \le x_i \le n$). All $x_i$ are distinct.
Print $q$ integers in a single line — the answers to the queries.
Input
The first line contains a single integer $n$ ($3 \le n \le 10^6$).
The second line contains $n - 1$ integers $p_2,p_3,\ldots,p_n$ ($1 \le p_i < i$) indicating that there is an edge between vertices $i$ and $p_i$. It is guaranteed that the given edges form a tree.
The third line contains a single integer $q$ ($1 \le q \le 10$).
The fourth line contains $q$ integers $x_1,x_2,\ldots,x_q$ ($1 \le x_i \le n$). All $x_i$ are distinct.
Output
Print $q$ integers in a single line — the answers to the queries.
Samples
4
1 2 2
4
1 2 3 4
1 2 2 2
7
1 2 3 4 2 1
7
2 1 3 7 5 6 4
3 3 4 5 5 5 4
3
1 2
1
1
1
Note
The graph in the first test after adding the edges:
