CF1637F Towers

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.

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 $ .