Score : $500$ points

问题描述

我们有一个包含 $N$ 个顶点和 $M$ 条边的连通无向图。
顶点编号从 $1$ 到 $N$，边编号从 $1$ 到 $M$。边 $i$ 连接顶点 $A_i$ 和 $B_i$。

高桥将从此图中移除零个或多个边。

当移除边 $i$ 时，若 $C_i \geq 0$，则会获得奖励 $C_i$；若 $C_i<0$，则会产生罚款 $|C_i|$。

求在移除边后图形必须保持连通的前提下，高桥可以获得的最大总奖励是多少。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a connected undirected graph with $N$ vertices and $M$ edges.
The vertices are numbered $1$ through $N$, and the edges are numbered $1$ through $M$. Edge $i$ connects Vertices $A_i$ and $B_i$.

Takahashi is going to remove zero or more edges from this graph.

When removing Edge $i$, a reward of $C_i$ is given if $C_i \geq 0$, and a fine of $|C_i|$ is incurred if $C_i<0$.

Find the maximum total reward that Takahashi can get when the graph must be connected after removing edges.

Constraints

• $2 \leq N \leq 2\times 10^5$
• $N-1 \leq M \leq 2\times 10^5$
• $1 \leq A_i,B_i \leq N$
• $-10^9 \leq C_i \leq 10^9$
• The given graph is connected.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$ $C_1$

$A_2$ $B_2$ $C_2$

$\vdots$

$A_M$ $B_M$ $C_M$

Output

Print the answer.

Sample Input 1

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

Sample Output 1

4

Removing Edges $4$ and $5$ yields a total reward of $4$. You cannot get any more, so the answer is $4$.

Sample Input 2

3 3
1 2 1
2 3 0
3 1 -1

Sample Output 2

1

There may be edges that give a negative reward when removed.

Sample Input 3

2 3
1 2 -1
1 2 2
1 1 3

Sample Output 3

5

There may be multi-edges and self-loops.

update @ 2024/3/10 09:39:06