Weighed Tree Radius
Description
You are given a tree of $n$ vertices and $n - 1$ edges. The $i$-th vertex has an initial weight $a_i$.
Let the distance $d_v(u)$ from vertex $v$ to vertex $u$ be the number of edges on the path from $v$ to $u$. Note that $d_v(u) = d_u(v)$ and $d_v(v) = 0$.
Let the weighted distance $w_v(u)$ from $v$ to $u$ be $w_v(u) = d_v(u) + a_u$. Note that $w_v(v) = a_v$ and $w_v(u) \neq w_u(v)$ if $a_u \neq a_v$.
Analogically to usual distance, let's define the eccentricity $e(v)$ of vertex $v$ as the greatest weighted distance from $v$ to any other vertex (including $v$ itself), or $e(v) = \max\limits_{1 \le u \le n}{w_v(u)}$.
Finally, let's define the radius $r$ of the tree as the minimum eccentricity of any vertex, or $r = \min\limits_{1 \le v \le n}{e(v)}$.
You need to perform $m$ queries of the following form:
- $v_j$ $x_j$ — assign $a_{v_j} = x_j$.
After performing each query, print the radius $r$ of the current tree.
The first line contains the single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of vertices in the tree.
The second line contains $n$ integers $a_1, \dots, a_n$ ($0 \le a_i \le 10^6$) — the initial weights of vertices.
Next $n - 1$ lines contain edges of tree. The $i$-th line contains two integers $u_i$ and $v_i$ ($1 \le u_i, v_i \le n$; $u_i \neq v_i$) — the corresponding edge. The given edges form a tree.
The next line contains the single integer $m$ ($1 \le m \le 10^5$) — the number of queries.
Next $m$ lines contain queries — one query per line. The $j$-th query contains two integers $v_j$ and $x_j$ ($1 \le v_j \le n$; $0 \le x_j \le 10^6$) — a vertex and it's new weight.
Print $m$ integers — the radius $r$ of the tree after performing each query.
Input
The first line contains the single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of vertices in the tree.
The second line contains $n$ integers $a_1, \dots, a_n$ ($0 \le a_i \le 10^6$) — the initial weights of vertices.
Next $n - 1$ lines contain edges of tree. The $i$-th line contains two integers $u_i$ and $v_i$ ($1 \le u_i, v_i \le n$; $u_i \neq v_i$) — the corresponding edge. The given edges form a tree.
The next line contains the single integer $m$ ($1 \le m \le 10^5$) — the number of queries.
Next $m$ lines contain queries — one query per line. The $j$-th query contains two integers $v_j$ and $x_j$ ($1 \le v_j \le n$; $0 \le x_j \le 10^6$) — a vertex and it's new weight.
Output
Print $m$ integers — the radius $r$ of the tree after performing each query.
6
1 3 3 7 0 1
2 1
1 3
1 4
5 4
4 6
5
4 7
4 0
2 5
5 10
5 5
7
4
5
10
7
Note
After the first query, you have the following tree:
The tree after the second query:
After the third query, the radius $r = e(2) = 5$:
