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.

Print one integer — the minimum required number of coins.

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

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 $ .