Fast Heavy-Light Decomposition

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Points: 10
Time limit: 2.0s
Memory limit: 256M

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Problem type

On an unweighted tree, there are N nodes and N - 1 edges. The nodes are numbered from 1 to N. All nodes i have a parent p_i, except for the root node, for which p_i is 0. All nodes are able to reach the root by repeatedly going up the parent of the current node.

Initially, all nodes have a value of 0. Perform the following operation on each node i:

Add x_i to the value of all nodes on the path from i to the root, including i and the root themselves.

Output the final values of all nodes.

Constraints

1 \leq N \leq 1\ 000\ 000

1 \leq x_i \leq 2\ 000

Input Specification

The first line contains one number: N.

The second line of input contains N numbers, the i-th number is p_i. If p_i is not 0, then node p_i is the parent of node i. If p_i is 0, then the node is the root. It is guaranteed that there are no cycles and exactly one node is the root.

The third line of input contains N numbers, the i-th number is x_i. This means you must add x_i to the value of all nodes from i to the root, inclusive.

Output Specification

Output N lines, the i-th line containing the value of node i.

Sample Input

5
0 1 1 3 3
1 2 4 8 16

Sample Output

31
2
28
8
16

Explanation for Sample

The path to root for 1 is just 1. So, 1 should be incremented by x_1 = 1.

The path to root for 2 is 2 \to 1. So, 2 and 1 should be incremented by x_2 = 2.

The path to root for 3 is 3 \to 1. So, 3 and 1 should be incremented by x_3 = 4.

The path to root for 4 is 4 \to 3 \to 1. So, 4, 3, and 1 should be incremented by x_4 = 8.

The path to root for 5 is 5 \to 3 \to 1. So, 5, 3, and 1 should be incremented by x_5 = 16.

Adding it all together the values at nodes 1, 2, 3, 4, and 5 are 31, 2, 28, 8, and 16, respectively.


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