OIer博客

给定一棵包含 $n$ 个点的树，第 $i$ 个点的高度为 $h_i$。你可以在这 $n$ 个点中建任意多个塔，对于每个塔，你都可以指定其所在的点的编号及其势能。建一个势能为 $e$ 的塔需要花费 $e$ 枚硬币（需要保证 $e>0$）。

对于一个树上的点 $x$，我们称其接收到了信号，当且仅当在树上存在两个建了塔的点 $u,v$（需要保证 $u\neq v$，但不需要保证 $x\neq u$ 或 $x\neq v$），使得 $x$ 在从 $u$ 到 $v$ 的路径上，且 $\min(e_u,e_v)\geqslant h_x$。

请求出最少需要花费多少枚硬币，才能使得树上所有点都能接受到信号。

数据范围：

– $2\leqslant n\leqslant 2\times 10^5$。
– $1\leqslant h_i\leqslant 10^9$。

Translated by Eason_AC

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.

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 $2$ .