A13758 | Boboniu and Jianghu

Since Boboniu finished building his Jianghu, he has been doing Kungfu on these mountains every day.

Boboniu designs a map for his $n$ mountains. He uses $n-1$ roads to connect all $n$ mountains. Every pair of mountains is connected via roads.

For the $i$ -th mountain, Boboniu estimated the tiredness of doing Kungfu on the top of it as $t_i$ . He also estimated the height of each mountain as $h_i$ .

A path is a sequence of mountains $M$ such that for each $i$ ( $1 \le i < |M|$ ), there exists a road between $M_i$ and $M_{i+1}$ . Boboniu would regard the path as a challenge if for each $i$ ( $1\le i<|M|$ ), $h_{M_i}\le h_{M_{i+1}}$ .

Boboniu wants to divide all $n-1$ roads into several challenges. Note that each road must appear in exactly one challenge, but a mountain may appear in several challenges.

Boboniu wants to minimize the total tiredness to do all the challenges. The tiredness of a challenge $M$ is the sum of tiredness of all mountains in it, i.e. $\sum_{i=1}^{|M|}t_{M_i}$ .

He asked you to find the minimum total tiredness. As a reward for your work, you'll become a guardian in his Jianghu.

The first line contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ), denoting the number of the mountains.

The second line contains $n$ integers $t_1, t_2, \ldots, t_n$ ( $1 \le t_i \le 10^6$ ), denoting the tiredness for Boboniu to do Kungfu on each mountain.

The third line contains $n$ integers $h_1, h_2, \ldots, h_n$ ( $1 \le h_i \le 10^6$ ), denoting the height of each mountain.

Each of the following $n - 1$ lines contains two integers $u_i$ , $v_i$ ( $1 \le u_i, v_i \le n, u_i \neq v_i$ ), denoting the ends of the road. It's guaranteed that all mountains are connected via roads.

Print one integer: the smallest sum of tiredness of all challenges.