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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System.Collections.Generic;
using System;

public class PriorityQueue<T> where T : IComparable
{
    private IComparer<T> _comparer = null;
    private int _type = 0;
    private T[] _heap;
    private int _sz = 0;
    private int _count = 0;
    /// <summary>
    /// Priority Queue with custom comparer
    /// </summary>
    public PriorityQueue(int maxSize, IComparer<T> comparer)
    {
        _heap = new T[maxSize];
        _comparer = comparer;
    }

    /// <summary>
    /// Priority queue
    /// </summary>
    /// <param name="maxSize">max size</param>
    /// <param name="type">0: asc, 1:desc</param>
    public PriorityQueue(int maxSize, int type = 0)
    {
        _heap = new T[maxSize];
        _type = type;
    }
    private int Compare(T x, T y)
    {
        if (_comparer != null) return _comparer.Compare(x, y);
        return _type == 0 ? x.CompareTo(y) : y.CompareTo(x);
    }
    public void Push(T x)
    {
        _count++;
        //node number
        var i = _sz++;
        while (i > 0)
        {
            //parent node number
            var p = (i - 1) / 2;

            if (Compare(_heap[p], x) <= 0) break;

            _heap[i] = _heap[p];
            i = p;
        }
        _heap[i] = x;
    }
    public T Pop()
    {
        _count--;
        T ret = _heap[0];
        T x = _heap[--_sz];

        int i = 0;
        while (i * 2 + 1 < _sz)
        {
            //children
            int a = i * 2 + 1;
            int b = i * 2 + 2;

            if (b < _sz && Compare(_heap[b], _heap[a]) < 0) a = b;

            if (Compare(_heap[a], x) >= 0) break;

            _heap[i] = _heap[a];
            i = a;
        }
        _heap[i] = x;
        return ret;
    }
    public int Count()
    {
        return _count;
    }
    public T Peek()
    {
        return _heap[0];
    }
    public bool Contains(T x)
    {
        for (int i = 0; i < _sz; i++) if (x.Equals(_heap[i])) return true;
        return false;
    }
    public void Clear()
    {
        while (this.Count() > 0) this.Pop();
    }
    public IEnumerator<T> GetEnumerator()
    {
        var ret = new List<T>();
        while (this.Count() > 0)
        {
            ret.Add(this.Pop());
        }
        foreach (var r in ret)
        {
            this.Push(r);
            yield return r;
        }
    }
    public T[] ToArray()
    {
        T[] array = new T[_sz];
        int i = 0;
        foreach (var r in this)
        {
            array[i++] = r;
        }
        return array;
    }
}
public struct Edge
{
    public int to { get; set; }
    public long d { get; set; }
}
public class Node : IComparable
{
    public int id { get; set; }
    public long d { get; set; }
    public int CompareTo(object obj)
    {
        var x = (Node)obj;
        if (this.d > x.d) return 1;
        else if (this.d == x.d) return 0;
        else return -1;
    }
}
public class Hello
{
    public const long INF = 999999999999999999;
    public static void Main()
    {
        var n = int.Parse(Console.ReadLine().Trim());
        var a = new List<Edge>[n];
        for (int i = 0; i < n; i++) a[i] = new List<Edge>();
        for (int i = 0; i < n - 1; i++)
        {
            string[] line = Console.ReadLine().Trim().Split(' ');
            var a1 = int.Parse(line[0]) - 1;
            var a2 = int.Parse(line[1]) - 1;
            var a3 = int.Parse(line[2]);
            a[a1].Add(new Edge { to = a2, d = a3 });
            a[a2].Add(new Edge { to = a1, d = a3 });
        }
        string[] line2 = Console.ReadLine().Trim().Split(' ');
        var q = int.Parse(line2[0]);
        var k = int.Parse(line2[1]) - 1;
        var mind = new long[n];
        for (int i = 0; i < n; i++) mind[i] = INF;
        goDijk(a, k, n, mind);
        for (int i = 0; i < q; i++)
        {
            line2 = Console.ReadLine().Trim().Split(' ');
            var x = int.Parse(line2[0]) - 1;
            var y = int.Parse(line2[1]) - 1;
            Console.WriteLine(mind[x] + mind[y]);
        }
    }
    public static void goDijk(List<Edge>[] aa, int r, int n, long[] mind)
    {
        var pq = new PriorityQueue<Node>(n + 10000, 0);
        pq.Push(new Node { id = r, d = 0 });
        while (pq.Count() > 0)
        {
            var targ = pq.Pop();
            var nowd = targ.d;
            var nowid = targ.id;
            if (mind[nowid] < nowd) continue;
            mind[nowid] = nowd;
            foreach (var x in aa[nowid])
                if (nowd + x.d < mind[x.to])
                    pq.Push(new Node { id = x.to, d = nowd + x.d });
        }
    }
}