Escape Through Leaf

有一颗 $n$ 个节点的树（节点从 $1$ 到 $n$ 依次编号），根节点为 $1$。每个节点有两个权值，第 $i$ 个节点的权值为 $a_i,b_i$。 你可以从一个节点跳到它的子树内任意一个节点上。从节点 $x$ 跳到节点 $y$ 一次的花费为 $a_x\times b_y$。跳跃多次走过一条路径的总费用为每次跳跃的费用之和。请分别计算出每个节点到达树的每个叶子节点的费用中的最小值。 注意：就算根节点的度数为 $1$，根节点也不算做叶子节点。另外，不能从一个节点跳到它自己. $2\leq n\leq 10^5$，$-10^5\leq a_i\leq 10^5$，$-10^5\leq b_i\leq 10^5$。

You are given a tree with $ n $ nodes (numbered from $ 1 $ to $ n $ ) rooted at node $ 1 $ . Also, each node has two values associated with it. The values for $ i $ -th node are $ a_{i} $ and $ b_{i} $ . You can jump from a node to any node in its subtree. The cost of one jump from node $ x $ to node $ y $ is the product of $ a_{x} $ and $ b_{y} $ . The total cost of a path formed by one or more jumps is sum of costs of individual jumps. For every node, calculate the minimum total cost to reach any leaf from that node. Pay attention, that root can never be leaf, even if it has degree $ 1 $ . Note that you cannot jump from a node to itself.

The first line of input contains an integer $ n $ ( $ 2<=n<=10^{5} $ ) — the number of nodes in the tree. The second line contains $ n $ space-separated integers $ a_{1},a_{2},...,a_{n}(-10^{5}<=a_{i}<=10^{5}) $ . The third line contains $ n $ space-separated integers $ b_{1},b_{2},...,b_{n}(-10^{5}<=b_{i}<=10^{5}) $ . Next $ n-1 $ lines contains two space-separated integers $ u_{i} $ and $ v_{i} $ ( $ 1<=u_{i},v_{i}<=n $ ) describing edge between nodes $ u_{i} $ and $ v_{i} $ in the tree.

Output $ n $ space-separated integers, $ i $ -th of which denotes the minimum cost of a path from node $ i $ to reach any leaf.

3
2 10 -1
7 -7 5
2 3
2 1

10 50 0 

4
5 -10 5 7
-8 -80 -3 -10
2 1
2 4
1 3

-300 100 0 0 

In the first example, node $ 3 $ is already a leaf, so the cost is $ 0 $ . For node $ 2 $ , jump to node $ 3 $ with cost $ a_{2}×b_{3}=50 $ . For node $ 1 $ , jump directly to node $ 3 $ with cost $ a_{1}×b_{3}=10 $ . In the second example, node $ 3 $ and node $ 4 $ are leaves, so the cost is $ 0 $ . For node $ 2 $ , jump to node $ 4 $ with cost $ a_{2}×b_{4}=100 $ . For node $ 1 $ , jump to node $ 2 $ with cost $ a_{1}×b_{2}=-400 $ followed by a jump from $ 2 $ to $ 4 $ with cost $ a_{2}×b_{4}=100 $ .