102714: [AtCoder]ABC271 E - Subsequence Path

Problem Statement

There are $N$ towns numbered $1, \dots, N$, and $M$ roads numbered $1, \dots, M$.
Every road is directed; road $i$ $(1 \leq i \leq M)$ leads you from Town $A_i$ to Town $B_i$. The length of road $i$ is $C_i$.

You are given a sequence $E = (E_1, \dots, E_K)$ of length $K$ consisting of integers between $1$ and $M$. A way of traveling from town $1$ to town $N$ using roads is called a good path if:

• the sequence of the roads' numbers arranged in the order used in the path is a subsequence of $E$.

Note that a subsequence of a sequence is a sequence obtained by removing $0$ or more elements from the original sequence and concatenating the remaining elements without changing the order.

Find the minimum sum of the lengths of the roads used in a good path.
If there is no good path, report that fact.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq M, K \leq 2 \times 10^5$
• $1 \leq A_i, B_i \leq N, A_i \neq B_i \, (1 \leq i \leq M)$
• $1 \leq C_i \leq 10^9 \, (1 \leq i \leq M)$
• $1 \leq E_i \leq M \, (1 \leq i \leq K)$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $K$
$A_1$ $B_1$ $C_1$
$\vdots$
$A_M$ $B_M$ $C_M$
$E_1$ $\ldots$ $E_K$

Output

Find the minimum sum of the lengths of the roads used in a good path.
If there is no good path, print -1.

Sample Input 1

3 4 4
1 2 2
2 3 2
1 3 3
1 3 5
4 2 1 2

Sample Output 1

4

There are two possible good paths as follows:

• Using road $4$. In this case, the sum of the length of the used road is $5$.
• Using road $1$ and $2$ in this order. In this case, the sum of the lengths of the used roads is $2 + 2 = 4$.

Therefore, the desired minimum value is $4$.

Sample Input 2

3 2 3
1 2 1
2 3 1
2 1 1

Sample Output 2

-1

There is no good path.

Sample Input 3

4 4 5
3 2 2
1 3 5
2 4 7
3 4 10
2 4 1 4 3

Sample Output 3

14

问题描述

有$N$个编号为$1, \dots, N$的城镇，以及$M$条编号为$1, \dots, M$的道路。
每条道路都是有向的；第$i$条道路$(1 \leq i \leq M)$将你从城镇$A_i$带到城镇$B_i$。第$i$条道路的长度为$C_i$。

给你一个长度为$K$的序列$E = (E_1, \dots, E_K)$，其中包含介于$1$和$M$之间的整数。从城镇$1$到城镇$N$使用道路的旅行方式称为一个好路径，如果：

• 按照路径中使用的顺序排列的道路的数字序列是序列$E$的子序列。

注意，序列的子序列是从原始序列中删除$0$个或多个元素并将剩余元素按顺序连接而获得的序列。

找到好路径中使用的道路长度之和的最小值。
如果没有好路径，请报告这一事实。

约束

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq M, K \leq 2 \times 10^5$
• $1 \leq A_i, B_i \leq N, A_i \neq B_i \, (1 \leq i \leq M)$
• $1 \leq C_i \leq 10^9 \, (1 \leq i \leq M)$
• $1 \leq E_i \leq M \, (1 \leq i \leq K)$
• 输入中的所有值都是整数。

输入

输入从标准输入按照以下格式给出：

$N$ $M$ $K$
$A_1$ $B_1$ $C_1$
$\vdots$
$A_M$ $B_M$ $C_M$
$E_1$ $\ldots$ $E_K$

输出

找到好路径中使用的道路长度之和的最小值。
如果没有好路径，请打印-1。

样例输入1

3 4 4
1 2 2
2 3 2
1 3 3
1 3 5
4 2 1 2

样例输出1

4

有两种可能的好路径如下：

• 使用道路$4$。在这种情况下，使用的道路长度之和为$5$。
• 按顺序使用道路$1$和$2$。在这种情况下，使用的道路长度之和为$2 + 2 = 4$。

因此，所需的最小值为$4$。

样例输入2

3 2 3
1 2 1
2 3 1
2 1 1

样例输出2

-1

没有好路径。

样例输入3

4 4 5
3 2 2
1 3 5
2 4 7
3 4 10
2 4 1 4 3

样例输出3

14