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ZOJ Problem Set - 4048
Red Black Tree

Time Limit: 3 Seconds      Memory Limit: 131072 KB

BaoBao has just found a rooted tree with $n$ vertices and $(n-1)$ weighted edges in his backyard. Among the vertices, $m$ of them are red, while the others are black. The root of the tree is vertex 1 and it's a red vertex.

Let's define the cost of a red vertex to be 0, and the cost of a black vertex to be the distance between this vertex and its nearest red ancestor.

Recall that

• The length of a path on the tree is the sum of the weights of the edges in this path.

• The distance between two vertices is the length of the shortest path on the tree to go from one vertex to the other.

• Vertex $u$ is the ancestor of vertex $v$ if it lies on the shortest path between vertex $v$ and the root of the tree (which is vertex 1 in this problem).

As BaoBao is bored, he decides to play $q$ games with the tree. For the $i$-th game, BaoBao will select $k_i$ vertices $v_{i, 1}, v_{i, 2}, \dots, v_{i, k_i}$ on the tree and try to minimize the maximum cost of these $k_i$ vertices by changing at most one vertex on the tree to a red vertex.

Note that

• BaoBao is free to change any vertex among all the $n$ vertices to a red vertex, NOT necessary among the $k_i$ vertiecs whose maximum cost he tries to minimize.

• All the $q$ games are independent. That is to say, the tree BaoBao plays with in each game is always the initial given tree, NOT the tree modified during the last game by changing at most one vertex.

Please help BaoBao calculate the smallest possible maximum cost of the given $k_i$ vertices in each game after changing at most one vertex to a red vertex.

#### Input

There are multiple test cases. The first line of the input is an integer $T$, indicating the number of test cases. For each test case:

The first line contains three integers $n$, $m$ and $q$ ($2 \le m \le n \le 10^5$, $1 \le q \le 2 \times 10^5$), indicating the size of the tree, the number of red vertices and the number of games.

The second line contains $m$ integers $r_1, r_2, \dots, r_m$ ($1 = r_1 < r_2 < \dots < r_m \le n$), indicating the red vertices.

The following $(n-1)$ lines each contains three integers $u_i$, $v_i$ and $w_i$ ($1 \le u_i, v_i \le n$, $1 \le w_i \le 10^9$), indicating an edge with weight $w_i$ connecting vertex $u_i$ and $v_i$ in the tree.

For the following $q$ lines, the $i$-th line will first contain an integer $k_i$ ($1 \le k_i \le n$). Then $k_i$ integers $v_{i, 1}, v_{i, 2}, \dots, v_{i, k_i}$ follow ($1 \le v_{i, 1} < v_{i, 2} < \dots < v_{i,k_i} \le n$), indicating the vertices whose maximum cost BaoBao has to minimize.

It's guaranteed that the sum of $n$ in all test cases will not exceed $10^6$, and the sum of $k_i$ in all test cases will not exceed $2 \times 10^6$.

#### Output

For each test case output $q$ lines each containing one integer, indicating the smallest possible maximum cost of the $k_i$ vertices given in each game after changing at most one vertex in the tree to a red vertex.

#### Sample Input

2
12 2 4
1 9
1 2 1
2 3 4
3 4 3
3 5 2
2 6 2
6 7 1
6 8 2
2 9 5
9 10 2
9 11 3
1 12 10
3 3 7 8
4 4 5 7 8
4 7 8 10 11
3 4 5 12
3 2 3
1 2
1 2 1
1 3 1
1 1
2 1 2
3 1 2 3


#### Sample Output

4
5
3
8
0
0
0


#### Hint

The first sample test case is shown above. Let's denote $C(v)$ as the cost of vertex $v$.

For the 1st game, the best choice is to make vertex 2 red, so that $C(3) = 4$, $C(7) = 3$ and $C(8) = 4$. So the answer is 4.

For the 2nd game, the best choice is to make vertex 3 red, so that $C(4) = 3$, $C(5) = 2$, $C(7) = 4$ and $C(8) = 5$. So the answer is 5.

For the 3rd game, the best choice is to make vertex 6 red, so that $C(7) = 1$, $C(8) = 2$, $C(10) = 2$ and $C(11) = 3$. So the answer is 3.

For the 4th game, the best choice is to make vertex 12 red, so that $C(4) = 8$, $C(5) = 7$ and $C(12) = 0$. So the answer is 8.

Author: WENG, Caizhi
Source: The 2018 ACM-ICPC Asia Qingdao Regional Contest, Online
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