200832: [AtCoder]ARC083 E - Bichrome Tree

Problem Statement

We have a tree with $N$ vertices. Vertex $1$ is the root of the tree, and the parent of Vertex $i$ ($2 \leq i \leq N$) is Vertex $P_i$.

To each vertex in the tree, Snuke will allocate a color, either black or white, and a non-negative integer weight.

Snuke has a favorite integer sequence, $X_1, X_2, ..., X_N$, so he wants to allocate colors and weights so that the following condition is satisfied for all $v$.

• The total weight of the vertices with the same color as $v$ among the vertices contained in the subtree whose root is $v$, is $X_v$.

Here, the subtree whose root is $v$ is the tree consisting of Vertex $v$ and all of its descendants.

Determine whether it is possible to allocate colors and weights in this way.

Constraints

• $1 \leq N \leq 1$ $000$
• $1 \leq P_i \leq i - 1$
• $0 \leq X_i \leq 5$ $000$

Inputs

Input is given from Standard Input in the following format:

$N$
$P_2$ $P_3$ $...$ $P_N$
$X_1$ $X_2$ $...$ $X_N$

Outputs

If it is possible to allocate colors and weights to the vertices so that the condition is satisfied, print POSSIBLE; otherwise, print IMPOSSIBLE.

Sample Input 1

3
1 1
4 3 2

Sample Output 1

POSSIBLE

For example, the following allocation satisfies the condition:

• Set the color of Vertex $1$ to white and its weight to $2$.
• Set the color of Vertex $2$ to black and its weight to $3$.
• Set the color of Vertex $3$ to white and its weight to $2$.

There are also other possible allocations.

Sample Input 2

3
1 2
1 2 3

Sample Output 2

IMPOSSIBLE

If the same color is allocated to Vertex $2$ and Vertex $3$, Vertex $2$ cannot be allocated a non-negative weight.

If different colors are allocated to Vertex $2$ and $3$, no matter which color is allocated to Vertex $1$, it cannot be allocated a non-negative weight.

Thus, there exists no allocation of colors and weights that satisfies the condition.

Sample Input 3

8
1 1 1 3 4 5 5
4 1 6 2 2 1 3 3

Sample Output 3

POSSIBLE

Sample Input 4

1

0

Sample Output 4

POSSIBLE