Ex - Painting Weighted Graph

AtCoder

IDabc235_h

Time3000ms

Memory256MB

Difficulty—

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: $N$ $M$ $K$ $A_1$ $B_1$ $C_1$ $\vdots$ $A_M$ $B_M$ $C_M$ [samples]

Samples

Input #1

3 2 1
1 2 1
2 3 2

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: ${},{1},{2},{3},{1,2},{1,2,3}$.

Input #2

5 0 2

Output #2

16

The given graph may not be connected.

Input #3

Output #3

40

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

API Response (JSON)

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      "content": "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 $C_i$.\nInitially, all vertices are painted black. You can do the fo...",
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