Score : $600$ points

问题描述

给定一个含 $N$ 个顶点和 $M$ 条边的无向图。第 $i$ 条边连接顶点 $A_i$ 和 $B_i$，且具有权重 $C_i$。

初始时，所有顶点都被涂成黑色。你可以最多进行 $K$ 次以下操作：

• 操作：选择任意顶点 $v$ 和任意整数 $x$。将从顶点 $v$ 出发，通过权重不大于 $x$ 的边可达的所有顶点（包括 $v$ 本身）涂成红色。

在完成操作后，有多少种可能的红色顶点集合？
请计算结果对 $998244353$ 取模后的数量。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Given is an undirected graph with $N$ vertices and $M$ edges. The $i$-th edge connects Vertices $A_i$ and $B_i$ and has a weight of $C_i$.

Initially, all vertices are painted black. You can do the following operation at most $K$ times.

• Operation: choose any vertex $v$ and any integer $x$. Paint red all vertices reachable from the vertex $v$ by traversing edges whose weights are at most $x$, including $v$ itself.

How many sets can be the set of vertices painted red after the operations?
Find the count modulo $998244353$.

Constraints

• $2 \leq N \leq 10^5$
• $0 \leq M \leq 10^5$
• $1 \leq K \leq 500$
• $1 \leq A_i,B_i \leq N$
• $1 \leq C_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

$A_1$ $B_1$ $C_1$

$\vdots$

$A_M$ $B_M$ $C_M$

Output

Print the answer.

Sample Input 1

3 2 1
1 2 1
2 3 2

Sample Output 1

6

For example, an operation with $(v,x)=(2,1)$ paints Vertices $1,2$ red, and an operation with $(v,x)=(1,0)$ paints Vertex $1$.

After at most one operation, the set of vertices painted red can be one of the following six: $\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,2,3\}$.

Sample Input 2

5 0 2

Sample Output 2

16

The given graph may not be connected.

Sample Input 3

6 8 2
1 2 1
2 3 2
3 4 3
4 5 1
5 6 2
6 1 3
1 2 10
1 1 100

Sample Output 3

40

The given graph may have multi-edges and self-loops.

update @ 2024/3/10 10:13:01