Boboniu and Jianghu
Description
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.
Input
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.
Output
Print one integer: the smallest sum of tiredness of all challenges.
Samples
5
40 10 30 50 20
2 3 2 3 1
1 2
1 3
2 4
2 5
160
5
1000000 1 1 1 1
1000000 1 1 1 1
1 2
1 3
1 4
1 5
4000004
10
510916 760492 684704 32545 484888 933975 116895 77095 127679 989957
402815 705067 705067 705067 623759 103335 749243 138306 138306 844737
1 2
3 2
4 3
1 5
6 4
6 7
8 7
8 9
9 10
6390572
Note
For the first example:
In the picture, the lighter a point is, the higher the mountain it represents. One of the best divisions is:
- Challenge $1$: $3 \to 1 \to 2$
- Challenge $2$: $5 \to 2 \to 4$
The total tiredness of Boboniu is $(30 + 40 + 10) + (20 + 10 + 50) = 160$. It can be shown that this is the minimum total tiredness.