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: Accepted)

module M = Set.Make(struct type t = int * int let compare = compare end)
let inf = 1010101010101010

let () =
  Scanf.scanf "%d" @@ fun n ->
  let g = Array.init (n+1) (fun _ -> []) in
  for _ = 1 to n-1 do
    Scanf.scanf " %d %d %d" @@ fun a b c ->
    g.(a) <- (b, c) :: g.(a);
    g.(b) <- (a, c) :: g.(b);
  done;

  Scanf.scanf " %d %d" @@ fun q k ->

  let dist = Array.make (n+1) inf in
  dist.(k) <- 0;

  let rec loop pqi =
    if M.is_empty pqi then () else
    let (d, u) = M.min_elt pqi in
    let pqi' = M.remove (d,u) pqi in

    if d > dist.(u) then loop pqi' else
      g.(u) |> List.fold_left (fun pq (v, c) ->
        if d+c < dist.(v) then begin
          dist.(v) <- d+c;
          M.add (d+c, v) pq
        end else pq) pqi'
    |> loop
  in loop (M.singleton (0, k));

  for _ = 1 to q do
    Scanf.scanf " %d %d" @@ fun x y ->
    dist.(x) + dist.(y) |> Printf.printf "%d\n"
  done