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 Rust (1.15.1) (judge verdict: Accepted)

fn main() {
  let n: usize = get::read();

  let edges: Vec<(usize, usize, isize)> = get::read_3tuples(n-1);
  let edges = edges.iter().map(|&(a,b,c)|(a-1,b-1,c)).collect::<Vec<_>>();

  let (q, k): (usize, usize) = get::read_tuple();
  let (q, k) = (q, k-1);

  let xys: Vec<(usize, usize)> = get::read_tuples(q);
  let xys = xys.iter().map(|&(a,b)|(a-1,b-1)).collect::<Vec<_>>();

  let g = Graph::new_labeled(n, &edges);
  let dist = g.dijkstra(k);

  let ans = xys.iter().map(|&(x,y)|dist[x]+dist[y]).collect::<Vec<_>>();
  
  put::print_vec(&ans, "\n");

}

struct Graph {
  size: usize,
  adj: Vec<Vec<(usize,isize)>>
}

impl Clone for Graph {
  fn clone(&self) -> Graph {
    Graph {
      size: self.size,
      adj: self.adj.clone()
    }
  }
}

#[allow(dead_code)]
impl Graph {
  fn new_nonlabeled(n: usize, edges: &[(usize,usize)]) -> Graph {
    let edges = edges.iter().map(|&(a,b)|(a,b,0isize)).collect::<Vec<_>>();
    Graph::new_labeled(n, &edges)
  }

  fn new_labeled(n: usize, edges: &[(usize,usize,isize)]) -> Graph {
    let mut g: Vec<Vec<(usize,isize)>> = vec![vec![]; n];
    for &(a, b, c) in edges {
      g[a].push((b,c));
      g[b].push((a,c));  // delete for digraph
    }
    Graph {size: n, adj: g}
  }

  fn is_connected(&self) -> bool {
    self.dfs(0).len() == self.size
  }

  fn size(&self) -> usize {
    self.size
  }

  fn dijkstra(&self, s: usize) -> Vec<isize> {
    use std::collections::BinaryHeap;
  
    let inf = isize::max_value();
    let mut dk: Vec<isize> = vec![inf; self.size];
    dk[s] = 0;
    
    let mut pq = BinaryHeap::new();
    pq.push((0,s));
    
    while !pq.is_empty() {
      assert!(pq.len()!=0);
      let (acc,v) = pq.pop().unwrap();
      let acc = -acc;
      for &(w,c) in &self.adj[v] {
        let cand = acc + c;
        if cand < dk[w] {
          dk[w] = cand;
          pq.push((-cand,w));
        }
      }
    }
    dk
  }

  fn warshall_floyd(&self) -> Vec<Vec<isize>> {
    let inf = isize::max_value() >> 1;
    let mut wf: Vec<Vec<isize>> = vec![vec![inf; self.size]; self.size];
    for i in 0 .. self.size {wf[i][i] = 0}
    
    for (i, next) in self.adj.iter().enumerate() {
      for &(j, x) in next {
        wf[i][j] = x
      }
    }
    for k in 0 .. self.size {
      for i in 0 .. self.size {
        for j in 0 .. self.size {
          wf[i][j] = std::cmp::min(wf[i][j], wf[i][k] + wf[k][j]);
        }
      }
    }
    wf
  }

  fn coloring2(&self) -> Option<(Vec<usize>, Vec<usize>)> {
    fn paint(v: usize, p: bool, g: &Graph, cv: &mut Vec<Option<bool>>) -> bool {
      match cv[v] {
        None => {
          let next = &g.adj[v];
          cv[v] = Some(p);
          next.iter().all(|&(w,_)|paint(w,!p,g,cv))},
        Some(q) => {q == p}
      }
    }

    let mut canvas: Vec<Option<bool>> = vec![None; self.size];
    let ans = paint(0, false, self, &mut canvas);
    let bs = canvas.iter().enumerate().filter(|&(_,&v)|v==Some(false)).map(|(i,_)|i).collect::<Vec<_>>();
    let ws = canvas.iter().enumerate().filter(|&(_,&v)|v==Some(true)).map(|(i,_)|i).collect::<Vec<_>>();
    if ans {Some((bs,ws))} else {None}
  }

  fn dfs(&self, s: usize) -> Vec<usize> {
    fn go(g: &Graph, current: usize, mut path: &mut Vec<usize>, mut visited: &mut Vec<bool>) {
      for &(next, _) in &g.adj[current] {
        if visited[next] {
          continue
        } else {
          visited[next] = true;
          path.push(next);
          go(&g, next, &mut path, &mut visited)
        }
      }
    }
    
    let mut path = vec![s];
    let mut visited = vec![false; self.adj.len()];
    visited[s] = true;
    go(&self, s, &mut path, &mut visited);
    path
  }
  
  fn cut(&mut self, v: usize, w: usize) {
    self.adj[v].retain(|&(t,_)| t != w);
    self.adj[w].retain(|&(t,_)| t != v);
  }
}

#[allow(dead_code)]
mod get {
  use std::io::*;
  use std::str::*;

  pub fn read<T: FromStr>() -> T {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    buf.trim().parse::<T>().ok().unwrap()
  }

  pub fn reads<T: FromStr>(n: usize) -> Vec<T> {
    let mut vec: Vec<T> = vec![];
    for _ in 0 .. n {
      vec.push(read());
    }
    vec
  }

  pub fn read_tuple<T1: FromStr, T2: FromStr>() -> (T1, T2) {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    let mut it = buf.trim().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    (x, y)
  }

  pub fn read_tuples<T1: FromStr, T2: FromStr>(n: usize) -> Vec<(T1, T2)> {
    let mut vec: Vec<(T1, T2)> = vec![];
    for _ in 0 .. n {
      vec.push(read_tuple());
    }
    vec
  }

  pub fn read_3tuple<T1: FromStr, T2: FromStr, T3: FromStr>() -> (T1, T2, T3) {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    let mut it = buf.trim().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    let z = it.next().unwrap().parse::<T3>().ok().unwrap();
    (x, y, z)
  }

  pub fn read_3tuples<T1: FromStr, T2: FromStr, T3: FromStr>(n: usize) -> Vec<(T1, T2, T3)> {
    let mut vec: Vec<(T1, T2, T3)> = vec![];
    for _ in 0 .. n {
      vec.push(read_3tuple());
    }
    vec
  }

  pub fn read_vec<T: FromStr>() -> Vec<T> {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    buf.trim().split_whitespace().map(|t| t.parse::<T>().ok().unwrap()).collect()
  }

  pub fn read_mat<T: FromStr>(h: usize) -> Vec<Vec<T>> {
    let mut mat: Vec<Vec<T>> = vec![];
    for _ in 0 .. h {
      mat.push(read_vec());
    }
    mat
  }
}

#[allow(dead_code)]
mod put {
  use std::string::*;

  pub fn print_vec<T: ToString>(vec: &Vec<T>, sep: &str) {
    let out = vec.iter().map(|e| e.to_string()).collect::<Vec<_>>().as_slice().join(sep);
    println!("{}", out);
  }

  pub fn print_mat<T: ToString>(mat: &Vec<Vec<T>>, sep: &str) {
    for v in mat {
      print_vec(v, sep);
    }
  }
}