Weighed Tree Radius

6000ms

524288K

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.

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.

Sample Input:

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

Sample Output:

Note:

After the first query, you have the following tree:

$\text{[math]}$ The marked vertex in the picture is the vertex with minimum $$$e(v)$$$, or $$$r = e(4) = 7$$$. The eccentricities of the other vertices are the following: $$$e(1) = 8$$$, $$$e(2) = 9$$$, $$$e(3) = 9$$$, $$$e(5) = 8$$$, $$$e(6) = 8$$$.

The tree after the second query:

$\text{[math]}$ The radius $$$r = e(1) = 4$$$.

After the third query, the radius $$$r = e(2) = 5$$$:

$\text{[math]}$

Informação

Provedor Codeforces

Código CF1783G