Nastya and Potions
Contest: Codeforces Round 888 (Div. 3)
Difficulty rating: 1500
Tags: dfs and similar, dp, graphs, sortings

Alchemist Nastya loves mixing potions. There are a total of $$$n$$$ types of potions, and one potion of type $$$i$$$ can be bought for $$$c_i$$$ coins.

Any kind of potions can be obtained in no more than one way, by mixing from several others. The potions used in the mixing process will be consumed. Moreover, no potion can be obtained from itself through one or more mixing processes.

As an experienced alchemist, Nastya has an unlimited supply of $$$k$$$ types of potions $$$p_1, p_2, \dots, p_k$$$, but she doesn't know which one she wants to obtain next. To decide, she asks you to find, for each $$$1 \le i \le n$$$, the minimum number of coins she needs to spend to obtain a potion of type $$$i$$$ next.

Input format:
The first line of each test contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

Each test case is described as follows:

The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k < n \le 2 \cdot 10^5$$$) — the total number of potion types and the number of potion types Nastya already has.

The second line contains $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$1 \le c_i \le 10^9$$$) — the costs of buying the potions.

The third line contains $$$k$$$ distinct integers $$$p_1, p_2, \dots, p_k$$$ ($$$1 \le p_i \le n$$$) — the indices of potions Nastya already has an unlimited supply of.

This is followed by $$$n$$$ lines describing ways to obtain potions by mixing.

Each line starts with the integer $$$m_i$$$ ($$$0 \le m_i < n$$$) — the number of potions required to mix a potion of the type $$$i$$$ ($$$1 \le i \le n$$$).

Then line contains $$$m_i$$$ distinct integers $$$e_1, e_2, \dots, e_{m_i}$$$ ($$$1 \le e_j \le n$$$, $$$e_j \ne i$$$) — the indices of potions needed to mix a potion of the type $$$i$$$. If this list is empty, then a potion of the type $$$i$$$ can only be bought.

It is guaranteed that no potion can be obtained from itself through one or more mixing processes.

It is guaranteed that the sum of all $$$n$$$ values across all test cases does not exceed $$$2 \cdot 10^5$$$. Similarly, it is guaranteed that the sum of all $$$m_i$$$ values across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output format:
For each test case, output $$$n$$$ integers — the minimum number of coins Nastya needs to spend to obtain a potion of each type.

Constraints:
Time limit: 3.0 s
Memory limit: 256.0 MB

Note:
In the first test case of the first sample, it is optimal:

- Get a potion of the first type by buying and mixing $$$2$$$, $$$4$$$ and $$$5$$$;
- a potion of the second type can only be obtained by purchasing it;
- Nastya already has an unlimited number of potions of the third type;
- a potion of the fourth type is more profitable to buy than to buy and mix other potions;
- a potion of the fifth type can only be obtained by purchasing it.

Example input:
4
5 1
30 8 3 5 10
3
3 2 4 5
0
0
2 3 5
0
3 2
5 143 3
1 3
1 2
0
2 1 2
5 1
5 4 1 3 4
2
2 4 5
3 3 5 4
2 1 4
1 5
0
4 2
1 1 5 4
2 4
3 2 4 3
0
2 2 4
1 2

Example output:
23 8 0 5 10 
0 143 0 
5 0 1 3 4 
0 0 0 0

Example input:
3
6 3
5 5 4 5 2 2
3 4 5
2 2 5
1 5
3 4 1 6
4 2 6 1 5
0
0
6 2
1 4 4 1 5 2
3 6
4 6 3 4 5
4 6 5 3 4
0
1 5
1 6
0
2 1
4 3
1
0
1 1

Example output:
0 0 0 0 0 2 
0 0 0 0 0 0 
0 0

Editorial:
To begin with, let's note that potions of types $$$p1, p2, \dots, p_k$$$ are essentially free, so we can replace their costs with $$$0$$$.
Let $$$ans[i]$$$ be the answer for the $$$i$$$-th potion. Each potion can be obtained in one of two ways: by buying it or by mixing it from other potions. For mixing, we obtain all the required potions at the minimum cost. That is, if there is a way to mix a potion of type $$$i$$$, then the answer for it is either $$$c_i$$$ or the sum of answers for all $$$e_i$$$. Since the graph in the problem does not have cycles, this can be done by a simple depth-first search.