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;
using System.Collections.Generic;
using System.Linq;

public class PriorityQueue<TKey, TValue>
{
    SortedDictionary<TKey, HashSet<TValue>> dict = new SortedDictionary<TKey, HashSet<TValue>>();
    Dictionary<TValue, TKey> rDict = new Dictionary<TValue, TKey>();
    int count = 0;

    public int Count
    {
        get
        {
            return count;
        }
    }

    public void Enqueue(TKey key, TValue value)
    {
        if (!dict.ContainsKey(key))
        {
            dict[key] = new HashSet<TValue>();
        }

        dict[key].Add(value);
        rDict[value] = key;
        count++;
    }

    public KeyValuePair<TKey, TValue> Dequeue()
    {
        var queue = dict.First();
        if (queue.Value.Count <= 1)
        {
            dict.Remove(queue.Key);
        }

        var val = queue.Value.First();
        queue.Value.Remove(val);
        count--;
        rDict.Remove(val);
        return new KeyValuePair<TKey, TValue>(queue.Key, val);
    }

    public KeyValuePair<TKey, TValue> Peek()
    {
        var queue = dict.First();
        var val = queue.Value.First();
        return new KeyValuePair<TKey, TValue>(queue.Key, val);
    }

    public void ChangePriority(TKey newKey, TValue value)
    {
        ChangePriority(rDict[value], newKey, value);
    }

    public void ChangePriority(TKey prevKey, TKey newKey, TValue value)
    {
        var set = dict[prevKey];
        set.Remove(value);
        if (set.Count == 0)
        {
            dict.Remove(prevKey);
        }
        count--;

        Enqueue(newKey, value);
    }

    public bool ContainsValue(TValue val)
    {
        return rDict.ContainsKey(val);
    }
}

class Solution
{
    static long[] Dijkstra(List<Tuple<int, int>>[] link, int k)
    {
        var n = link.Length - 1;
        var d = Enumerable.Repeat(long.MaxValue, n + 1).ToArray();
        d[k] = 0;
        var q = new PriorityQueue<long, int>();
        q.Enqueue(0, k);

        while (q.Count > 0)
        {
            var u = q.Dequeue();
            foreach (var nei in link[u.Value])
            {
                var v = nei.Item1;
                var len = nei.Item2;
                if (d[v] > d[u.Value] + len)
                {
                    d[v] = d[u.Value] + len;
                    if (q.ContainsValue(v))
                    {
                        q.ChangePriority(d[v], v);
                    }
                    else
                    {
                        q.Enqueue(d[v], v);
                    }
                }
            }
        }

        return d;
    }

    static void Main()
    {
        var n = int.Parse(Console.ReadLine());
        var link = Enumerable.Range(0, n + 1).Select(_ => new List<Tuple<int, int>>()).ToArray();
        for (int i = 0; i < n - 1; i++)
        {
            var v = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
            link[v[0]].Add(Tuple.Create(v[1], v[2]));
            link[v[1]].Add(Tuple.Create(v[0], v[2]));
        }

        var vals = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        var q = vals[0];
        var k = vals[1];
        var query = new int[q][];
        for (int i = 0; i < q; i++)
        {
            query[i] = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        }

        var d = Dijkstra(link, k);

        foreach (var que in query)
        {
            Console.WriteLine($"{d[que[0]] + d[que[1]]}");
        }
    }
}