A13698 · Avoid Rainbow Cycles

题目描述

You are given $m$ sets of integers $A_1, A_2, \ldots, A_m$ ; elements of these sets are integers between $1$ and $n$ , inclusive. There are two arrays of positive integers $a_1, a_2, \ldots, a_m$ and $b_1, b_2, \ldots, b_n$ . In one operation you can delete an element $j$ from the set $A_i$ and pay $a_i + b_j$ coins for that. You can make several (maybe none) operations (some sets can become empty). After that, you will make an edge-colored undirected graph consisting of $n$ vertices. For each set $A_i$ you will add an edge $(x, y)$ with color $i$ for all $x, y \in A_i$ and $x < y$ . Some pairs of vertices can be connected with more than one edge, but such edges have different colors. You call a cycle $i_1 \to e_1 \to i_2 \to e_2 \to \ldots \to i_k \to e_k \to i_1$ ( $e_j$ is some edge connecting vertices $i_j$ and $i_{j+1}$ in this graph) rainbow if all edges on it have different colors. Find the minimum number of coins you should pay to get a graph without rainbow cycles.

The first line contains two integers $m$ and $n$ ( $1 \leq m, n \leq 10^5$ ), the number of sets and the number of vertices in the graph. The second line contains $m$ integers $a_1, a_2, \ldots, a_m$ ( $1 \leq a_i \leq 10^9$ ). The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ( $1 \leq b_i \leq 10^9$ ). In the each of the next of $m$ lines there are descriptions of sets. In the $i$ -th line the first integer $s_i$ ( $1 \leq s_i \leq n$ ) is equal to the size of $A_i$ . Then $s_i$ integers follow: the elements of the set $A_i$ . These integers are from $1$ to $n$ and distinct. It is guaranteed that the sum of $s_i$ for all $1 \leq i \leq m$ does not exceed $2 \cdot 10^5$ .

Print one integer: the minimum number of coins you should pay for operations to avoid rainbow cycles in the obtained graph.

输入 #1

输出 #1

输入 #2

7 8
3 6 7 9 10 7 239
8 1 9 7 10 2 6 239
3 2 1 3
2 4 1
3 1 3 7
2 4 3
5 3 4 5 6 7
2 5 7
1 8

输出 #2