Problem:
Score : 400 points

Problem Statement

You are given a tree with N vertices.

Here, a tree is a kind of graph, and more specifically, a connected undirected graph with N-1 edges, where N is the number of its vertices.

The i-th edge (1≤i≤N-1) connects Vertices a_i and b_i, and has a length of c_i.

You are also given Q queries and an integer K. In the j-th query (1≤j≤Q):

find the length of the shortest path from Vertex x_j and Vertex y_j via Vertex K.

Constraints

3≤N≤10^5

1≤a_i,b_i≤N (1≤i≤N-1)

1≤c_i≤10^9 (1≤i≤N-1)

The given graph is a tree.

1≤Q≤10^5

1≤K≤N

1≤x_j,y_j≤N (1≤j≤Q)

x_j≠y_j (1≤j≤Q)

x_j≠K,y_j≠K (1≤j≤Q)

Input

Input is given from Standard Input in the following format:

N
a_1 b_1 c_1
:
a_{N-1} b_{N-1} c_{N-1}
Q K
x_1 y_1
:
x_{Q} y_{Q}

Output

Print the responses to the queries in Q lines.

In the j-th line j(1≤j≤Q), print the response to the j-th query.

Sample Input 1

5
1 2 1
1 3 1
2 4 1
3 5 1
3 1
2 4
2 3
4 5

Sample Output 1

3
2
4

The shortest paths for the three queries are as follows:

Query 1: Vertex 2 → Vertex 1 → Vertex 2 → Vertex 4 : Length 1+1+1=3

Query 2: Vertex 2 → Vertex 1 → Vertex 3 : Length 1+1=2

Query 3: Vertex 4 → Vertex 2 → Vertex 1 → Vertex 3 → Vertex 5 : Length 1+1+1+1=4

Sample Input 2

7
1 2 1
1 3 3
1 4 5
1 5 7
1 6 9
1 7 11
3 2
1 3
4 5
6 7

Sample Output 2

5
14
22

The path for each query must pass Vertex K=2.

Sample Input 3

10
1 2 1000000000
2 3 1000000000
3 4 1000000000
4 5 1000000000
5 6 1000000000
6 7 1000000000
7 8 1000000000
8 9 1000000000
9 10 1000000000
1 1
9 10

Sample Output 3

17000000000

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

module MakeBinaryHeap (M : sig type t val compare : t -> t -> int end) = struct

  type t = { node : M.t array; mutable size : int }

  let make n (init : M.t) = { node = Array.make n init; size = 0 }

  let empty_p t = if t.size = 0 then true else false

  let pop t =
    if t.size <= 0 then failwith "out of size" else
    let rec max_heapify i =
      let l = 2*i in
      let r = 2*i + 1 in
      let m = if l <= t.size && M.compare t.node.(l) t.node.(i) > 0 then l else i in
      let m = if r <= t.size && M.compare t.node.(r) t.node.(m) > 0 then r else m in
      if m = i then ()
      else begin
        let tmp = t.node.(i) in
        t.node.(i) <- t.node.(m);
        t.node.(m) <- tmp;
        max_heapify m
      end in
    let ret = t.node.(1) in
    t.node.(1) <- t.node.(t.size);
    t.size <- t.size - 1;
    max_heapify 1;
    ret

  let push x t =
    let parent i = int_of_float (floor (float_of_int i) /. 2.) in
    let rec doit i =
      let p = parent i in
      if i <= 1 || M.compare t.node.(p) t.node.(i) >= 0 then ()
      else begin
        let tmp = t.node.(i) in
        t.node.(i) <- t.node.(p);
        t.node.(p) <- tmp;
        doit p
      end in
    t.size <- t.size + 1;
    t.node.(t.size) <- x;
    doit t.size

end

module H = MakeBinaryHeap(struct type t = int * int let compare (_, x) (_, y) = y - x end)

let dijkstra t n k =
  let reached_p = Array.make n false in
  let d = Array.make n max_int in
  d.(k) <- 0;
  let h = H.make 10000 (0, 0) in
  H.push (k, 0) h;
  while not (H.empty_p h) do
    let (u, x) = H.pop h in
    reached_p.(u) <- true;
    if x > d.(u) then ()
    else
      Array.iteri (fun v c ->
        if c = max_int then ()
        else begin
          let x = d.(u) + c in
          if not reached_p.(v) && x < d.(v) then begin
            d.(v) <- x;
            H.push (v, x) h;
          end
        end) t.(u);
  done;
  d

let () =
  let n = Scanf.scanf "%d " (fun i -> i) in
  let t = Array.make_matrix n n max_int in
  for _ = 1 to n - 1 do
    let a, b, c = Scanf.scanf "%d %d %d " (fun a b c -> a, b, c) in
    let a, b = a - 1, b - 1 in
    t.(a).(b) <- c;
    t.(b).(a) <- c;
  done;
  let q, k = Scanf.scanf "%d %d " (fun q k -> q, k) in
  let k = k - 1 in
  let d = dijkstra t n k in
  for _ = 0 to q - 1 do
    let x, y = Scanf.scanf "%d %d " (fun x y -> x, y) in
    let x, y = x - 1, y - 1 in
    Printf.printf "%d\n" (d.(x) + d.(y))
  done