Problem:
Score : 400 points

Problem Statement

Given is a rooted tree with N vertices numbered 1 to N.
The root is Vertex 1, and the i-th edge (1 \leq i \leq N - 1) connects Vertex a_i and b_i.

Each of the vertices has a counter installed. Initially, the counters on all the vertices have the value 0.

Now, the following Q operations will be performed:

Operation j (1 \leq j \leq Q): Increment by x_j the counter on every vertex contained in the subtree rooted at Vertex p_j.

Find the value of the counter on each vertex after all operations.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq Q \leq 2 \times 10^5

1 \leq a_i < b_i \leq N

1 \leq p_j \leq N

1 \leq x_j \leq 10^4

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N Q
a_1 b_1
:
a_{N-1} b_{N-1}
p_1 x_1
:
p_Q x_Q

Output

Print the values of the counters on Vertex 1, 2, \ldots, N after all operations, in this order, with spaces in between.

Sample Input 1

4 3
1 2
2 3
2 4
2 10
1 100
3 1

Sample Output 1

100 110 111 110

The tree in this input is as follows:

Each operation changes the values of the counters on the vertices as follows:

Operation 1: Increment by 10 the counter on every vertex contained in the subtree rooted at Vertex 2, that is, Vertex 2, 3, 4. The values of the counters on Vertex 1, 2, 3, 4 are now 0, 10, 10, 10, respectively.

Operation 2: Increment by 100 the counter on every vertex contained in the subtree rooted at Vertex 1, that is, Vertex 1, 2, 3, 4. The values of the counters on Vertex 1, 2, 3, 4 are now 100, 110, 110, 110, respectively.

Operation 3: Increment by 1 the counter on every vertex contained in the subtree rooted at Vertex 3, that is, Vertex 3. The values of the counters on Vertex 1, 2, 3, 4 are now 100, 110, 111, 110, respectively.

Sample Input 2

6 2
1 2
1 3
2 4
3 6
2 5
1 10
1 10

Sample Output 2

20 20 20 20 20 20

Solution in OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

open Batteries
module EnumL = Enum.Labels
module ListL = List.Labels
module ArrayL = Array.Labels

let dbg0 x = Printf.eprintf "[debug]%s\n" @@ dump x
let dbg1 x = dbg0 x; x

let id = identity

let (++) n m = List.range n `To m
let (++^) n m = if n < m then List.range n `To (pred m) else []

let scan fmt = Scanf.sscanf (read_line ()) fmt

let scan_lines n fmt f =
  if n = 0 then Enum.empty ()
  else
    List.map (fun _ -> scan fmt f) (List.range 1 `To n)
    |> List.enum

let scan_list cnv =
  String.split_on_char ' ' @@ read_line ()
  |> List.map cnv

let scan_array cnv = Array.of_list @@ scan_list cnv

let scan_matrix n m e conv =
  let arr = Array.make_matrix n m e in
  ListL.iter (List.range 0 `To (pred n))
    ~f:(fun i -> arr.(i) <- Array.of_list @@ scan_list conv);
  arr

let rec powerset e =
  match Enum.get e with
  | None -> Enum.singleton @@ Enum.empty ()
  | Some v ->
    let f = powerset e in
    let g = Enum.clone f in
    EnumL.map f ~f:(fun x -> let y = Enum.clone x in push y v; y)
    |>  Enum.append g

let permutations l =
  let rec aux l =
    let rec interleave x = function
      | [] -> [[x]]
      | (hd::tl) as lst ->
        (x::lst) ::
        (ListL.map ~f:(List.cons hd)  @@ interleave x tl)
    in
    match l with
    | [] -> [[]]
    | hd::tl -> List.concat @@ List.map (interleave hd) @@ aux tl in
  let l = List.sort (List.compare Int.compare) @@ aux @@ List.of_enum l in
  List.enum % ListL.map ~f:List.enum @@ l

let intersection l =
  EnumL.filter ~f:(fun x -> EnumL.exists ~f:((=) x) @@ Enum.clone l)

let lower_bound n f =
  let rec aux l u =
    if u - l > 1 then
      let m = (l + u) / 2 in
      if f m then aux l m
      else aux m u
    else u in aux (-1) n

let (n, q) = scan "%d %d" Tuple.Tuple2.make
let ab = List.of_enum @@ scan_lines (n-1) "%d %d" Tuple.Tuple2.make
let px = List.of_enum @@ scan_lines q "%d %d" Tuple.Tuple2.make

let () =
  let tree = Array.init n (fun _ -> []) in
  let ntree = Array.make n 0 in
  let rec cut p  =
    let cs = tree.(p) in
    ListL.iter cs
      ~f:(fun c ->
          tree.(c) <- ListL.filter tree.(c) ~f:((<>) p);
          cut c)
  in
  ListL.iter ab ~f:(fun (a, b) ->
      tree.(pred a) <- (pred b)::tree.(pred a);
      tree.(pred b) <- (pred a)::tree.(pred b));
  cut 0;
  ListL.iter px ~f:(fun (p,x) ->
      let rec aux x p =
        ntree.(p) <- ntree.(p) + x;
        let cs = tree.(p) in
        ListL.iter cs ~f:(aux x)
      in
      aux x (pred p));
  IO.to_string (Array.print ~first:"" ~last:"\n" ~sep:" " Int.print) ntree
  |> Printf.printf "%s"