100703: [AtCoder]ABC070 D - Transit Tree Path

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