Foreign Friends

AtCoder

IDabc245_g

Time2000ms

Memory256MB

Difficulty—

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: $N$ $M$ $K$ $L$ $A_1$ $A_2$ $\cdots$ $A_N$ $B_1$ $B_2$ $\cdots$ $B_L$ $U_1$ $V_1$ $C_1$ $U_2$ $V_2$ $C_2$ $\vdots$ $U_M$ $V_M$ $C_M$ [samples]

Samples

Input #1

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 $15+10=25$.

Input #2

3 1 3 1
1 2 3
1
1 2 1000000000

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.

API Response (JSON)

{
  "problem": {
    "name": "Foreign Friends",
    "description": {
      "content": "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 natio",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc245_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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.  \nEach person belongs to exactly one natio...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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