[ICPC2024 Xi'an I] Fix the Tree

You are given a tree consisting of $n$ vertices. Each vertex $i$ of this tree has a value $w_i$ assigned to it. Now the vertex $u$ will be broken. Once it's broken, vertex $u$ and all edges with one end at vertex $u$ will be removed from the tree. To make the tree connected, you can do the following operation any number of times(possibly zero) in any order: - First choose two vertices $u$ and $v$ from the tree; - Then pay $w_u+w_v$ coins, and add an edge between vertices $u$ and $v$; - At last, replace $w_u+1$ with $w_u$, replace $w_v+1$ with $w_v$. Your task is to calculate the minimum number of coins to be paid. But you don't know which vertex $u$ is, so you need to find the answer for each $1\le u\le n$. Please answer all the queries independently.

The first line contains a single integer $n(2\le n\le 10^6)$ --- the number of vertices in this tree. The next line contains $n$ numbers, the $i$ -th number is $w_i(1\le w_i\le n)$. The next $n-1$ lines contain a description of the tree's edges. The $i$ -th of these lines contains two integers $u_i$ and $v_i(1\le u_i,v_i\le n) $ --- the numbers of vertices connected by the $i$ -th edge. It is guaranteed that the given edges form a tree.

Print $n$ integers, the $i$ -th integer is the answer when $u=i$.

6
1 1 1 1 1 1
1 2
1 3
1 4
2 5
2 6

4 4 0 0 0 0

4
1 2 3 4
1 2
1 3
1 4

12 0 0 0

7
1 2 3 4 5 6 7
1 2
1 3
2 4
2 5
3 6
3 7

5 12 16 0 0 0 0

给定一个有 $n$ 个点组成的树，每个点有一个权值 $w_i$。 点 $u$ 和相邻的边被删除。 你可以进行以下操作任意次数（可以为 $0$）,让树重新成为连通图： 1. 选择两个点 $u$、$v$； 2. 花费 $w_u + w_v$ 的代价连接一条边 $(u,v)$； 3. $w_u \leftarrow w_u+1,w_v \leftarrow w_v+1$。 对于每个 $u$ 计算最小代价。