Escape Through Leaf

3000ms

262144K

Description:

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.

Input:

The first line of input contains an integer n (2 ≤ n ≤ 10⁵) — the number of nodes in the tree.

The second line contains n space-separated integers a₁, a₂, ..., a_n( - 10⁵ ≤ a_i ≤ 10⁵).

The third line contains n space-separated integers b₁, b₂, ..., b_n( - 10⁵ ≤ b_i ≤ 10⁵).

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:

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

Sample Input:

Sample Output:

10 50 0

Sample Input:

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

Sample Output:

-300 100 0 0

Note:

In the first example, node 3 is already a leaf, so the cost is 0. For node 2, jump to node 3 with cost a₂ × b₃ = 50. For node 1, jump directly to node 3 with cost a₁ × b₃ = 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 a₂ × b₄ = 100. For node 1, jump to node 2 with cost a₁ × b₂ = - 400 followed by a jump from 2 to 4 with cost a₂ × b₄ = 100.

Informação

Provedor Codeforces

Código CF932F