Score : $600$ points

问题陈述

设有 $N$ 名人物和 $K$ 个国家，分别标记为人物 $1$, 人物 $2$, $\ldots$, 人物 $N$ 和国家 $1$, 国家 $2$, $\ldots$, 国家 $K$。 每个人物恰好属于一个国家：人物 $i$ 属于国家 $A_i$。此外，在这 $N$ 名人物中有 $L$ 名受欢迎的人物：人物 $B_1$, 人物 $B_2$, $\ldots$, 人物 $B_L$ 是受欢迎的。最初，这 $N$ 名人物中任意两人之间都不是朋友关系。

对于 $M$ 对人物，神明 Takahashi 可以通过支付一定的代价使他们成为朋友：对于每个 $1\leq i\leq M$，他可以支付 $C_i$ 的代价，使得人物 $U_i$ 和人物 $V_i$ 成为朋友。

现在，对于每 $1\leq i\leq N$，解决以下问题。

Takahashi 是否能够以最小的总代价使得人物 $i$ 间接与一个来自不同国家的受欢迎人物建立朋友关系？如果可以做到，请找出所需的最小总成本。这里，当存在非负整数 $n$ 和人物序列 $(u_0, u_1, \ldots, u_n)$ 且满足 $u_0=s$、$u_n=t$ 以及人物 $u_i$ 和人物 $u_{i+1}$ 在 $0\leq i < n$ 范围内都是朋友时，称人物 $s$ 是人物 $t$ 的间接朋友。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ people and $K$ nations, labeled as Person $1$, Person $2$, $\ldots$, Person $N$ and Nation $1$, Nation $2$, $\ldots$, Nation $K$, respectively.
Each person belongs to exactly one nation: Person $i$ belongs to Nation $A_i$. Additionally, there are $L$ popular people among them: Person $B_1$, Person $B_2$, $\ldots$, Person $B_L$ are popular. Initially, no two of the $N$ people are friends.

For $M$ pairs of people, Takahashi, a God, can make them friends by paying a certain cost: for each $1\leq i\leq M$, he can pay the cost of $C_i$ to make Person $U_i$ and Person $V_i$ friends.

Now, for each $1\leq i\leq N$, solve the following problem.

Can Takahashi make Person $i$ an indirect friend of a popular person belonging to a nation different from that of Person $i$? If he can do so, find the minimum total cost needed. Here, Person $s$ is said to be an indirect friend of Person $t$ when there exists a non-negative integer $n$ and a sequence of people $(u_0, u_1, \ldots, u_n)$ such that $u_0=s$, $u_n=t$, and Person $u_i$ and Person $u_{i+1}$ are friends for each $0\leq i < n$.

Constraints

• $2 \leq N \leq 10^5$
• $1 \leq M \leq 10^5$
• $1 \leq K \leq 10^5$
• $1 \leq L \leq N$
• $1 \leq A_i \leq K$
• $1 \leq B_1<B_2<\cdots<B_L\leq N$
• $1\leq C_i\leq 10^9$
• $1\leq U_i<V_i\leq N$
• $(U_i, V_i)\neq (U_j,V_j)$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$ $L$

$A_1$ $A_2$ $\cdots$ $A_N$

$B_1$ $B_2$ $\cdots$ $B_L$

$U_1$ $V_1$ $C_1$

$U_2$ $V_2$ $C_2$

$\vdots$

$U_M$ $V_M$ $C_M$

Output

Let $X_i$ defined as follows: $X_i$ is $-1$ if it is impossible to make Person $i$ an indirect friend of a popular person belonging to a nation different from that of Person $i$; otherwise, $X_i$ is the minimum total cost needed to do so. Print $X_1, X_2, \ldots, X_N$ in one line, with spaces in between.

Sample Input 1

4 4 2 2
1 1 2 2
2 3
1 2 15
2 3 30
3 4 40
1 4 10

Sample Output 1

45 30 30 25

Person $1$, $2$, $3$, $4$ belong to Nation $1$, $1$, $2$, $2$, respectively, and there are two popular people: Person $2$ and $3$. Here,

• For Person $1$, the only popular person belonging to a different nation is Person $3$. To make them indirect friends with the minimum cost, we should pay the cost of $15$ to make Person $1$ and $2$ friends and pay $30$ to make Person $2$ and $3$ friends, for a total of $15+30=45$.
• For Person $2$, the only popular person belonging to a different nation is Person $3$. The minimum cost is achieved by making Person $2$ and $3$ friends by paying $30$.
• For Person $3$, the only popular person belonging to a different nation is Person $2$. The minimum cost is achieved by making Person $2$ and $3$ friends by paying $30$.
• For Person $4$, the only popular person belonging to a different nation is Person $2$. To make them indirect friends with the minimum cost, we should pay the cost of $15$ to make Person $1$ and $2$ friends and pay $10$ to make Person $1$ and $4$ friends, for a total of $15+10=25$.

Sample Input 2

3 1 3 1
1 2 3
1
1 2 1000000000

Sample Output 2

-1 1000000000 -1

Note that, for Person $1$, Person $1$ itself is indeed an indirect friend, but it does not belong to a different nation, so there is no popular person belonging to a different nation.

update @ 2024/3/10 10:32:32