Problem:
Score : 700 points

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

Solution in OCaml (4.02.3) (judge verdict: Accepted)

module IntMap = Map.Make (struct
  type t = int
  let compare = compare
end)

let () = Scanf.scanf "%d\n" @@ fun n ->
  let es = Array.make n [] in
  let xs =
    if n = 1 then
      Scanf.scanf "\n%d\n" @@ Array.make 1
    else begin
      for i = 1 to n - 1 do
        Scanf.scanf "%d " @@ fun p ->
          es.(p - 1) <- i :: es.(p - 1)
      done;
      Array.init n @@ fun _ -> Scanf.scanf "%d " @@ fun x -> x
    end in
  let rec solve v =
    let xys = List.map (fun u -> xs.(u), solve u) es.(v) in
    let most = List.fold_right (fun (x, y) -> ( + ) @@ max x y) xys 0 in
    let least = List.fold_right (fun (x, y) -> ( + ) @@ min x y) xys 0 in
    if xs.(v) < least then raise Not_found
    else IntMap.fold (fun _ -> min)
      (List.fold_left (fun m (w, v_) ->
        let le, _, _ = IntMap.split (xs.(v) - least - w + 1) m in
        IntMap.fold (fun w' v' m ->
          IntMap.add (w + w')
            (max (v_ + v') @@
              try IntMap.find (w + w') m
              with Not_found -> min_int) m) le m) (IntMap.singleton 0 most) @@
        List.map (fun (x, y) -> abs (x - y), ~- (abs (x - y))) xys) max_int in
  print_endline @@
  try ignore (solve 0); "POSSIBLE"
  with Not_found -> "IMPOSSIBLE"