Towers
Description
You are given a tree with $n$ vertices numbered from $1$ to $n$. The height of the $i$-th vertex is $h_i$. You can place any number of towers into vertices, for each tower you can choose which vertex to put it in, as well as choose its efficiency. Setting up a tower with efficiency $e$ costs $e$ coins, where $e > 0$.
It is considered that a vertex $x$ gets a signal if for some pair of towers at the vertices $u$ and $v$ ($u \neq v$, but it is allowed that $x = u$ or $x = v$) with efficiencies $e_u$ and $e_v$, respectively, it is satisfied that $\min(e_u, e_v) \geq h_x$ and $x$ lies on the path between $u$ and $v$.
Find the minimum number of coins required to set up towers so that you can get a signal at all vertices.
The first line contains an integer $n$ ($2 \le n \le 200\,000$) — the number of vertices in the tree.
The second line contains $n$ integers $h_i$ ($1 \le h_i \le 10^9$) — the heights of the vertices.
Each of the next $n - 1$ lines contain a pair of numbers $v_i, u_i$ ($1 \le v_i, u_i \le n$) — an edge of the tree. It is guaranteed that the given edges form a tree.
Print one integer — the minimum required number of coins.
Input
The first line contains an integer $n$ ($2 \le n \le 200\,000$) — the number of vertices in the tree.
The second line contains $n$ integers $h_i$ ($1 \le h_i \le 10^9$) — the heights of the vertices.
Each of the next $n - 1$ lines contain a pair of numbers $v_i, u_i$ ($1 \le v_i, u_i \le n$) — an edge of the tree. It is guaranteed that the given edges form a tree.
Output
Print one integer — the minimum required number of coins.
Samples
3
1 2 1
1 2
2 3
4
5
1 3 3 1 3
1 3
5 4
4 3
2 3
7
2
6 1
1 2
12
Note
In the first test case it's optimal to install two towers with efficiencies $2$ at vertices $1$ and $3$.
In the second test case it's optimal to install a tower with efficiency $1$ at vertex $1$ and two towers with efficiencies $3$ at vertices $2$ and $5$.
In the third test case it's optimal to install two towers with efficiencies $6$ at vertices $1$ and $2$.