Teleporting Takahashi 2

AtCoder

IDabc372_f

Time3000ms

Memory256MB

Difficulty—

There is a simple directed graph $G$ with $N$ vertices and $N+M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $N+M$. Edge $i$ $(1 \leq i \leq N)$ goes from vertex $i$ to vertex $i+1$. (Here, vertex $N+1$ is considered as vertex $1$.) Edge $N+i$ $(1 \leq i \leq M)$ goes from vertex $X_i$ to vertex $Y_i$. Takahashi is at vertex $1$. At each vertex, he can move to any vertex to which there is an outgoing edge from the current vertex. Compute the number of ways he can move exactly $K$ times. That is, find the number of integer sequences $(v_0, v_1, \dots, v_K)$ of length $K+1$ satisfying all of the following three conditions: * $1 \leq v_i \leq N$ for $i = 0, 1, \dots, K$. * $v_0 = 1$. * There is a directed edge from vertex $v_{i-1}$ to vertex $v_i$ for $i = 1, 2, \ldots, K$. Since this number can be very large, print it modulo $998244353$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $0 \leq M \leq 50$ * $1 \leq K \leq 2 \times 10^5$ * $1 \leq X_i, Y_i \leq N$, $X_i \neq Y_i$ * All of the $N+M$ directed edges are distinct. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $K$ $X_1$ $Y_1$ $X_2$ $Y_2$ $\vdots$ $X_M$ $Y_M$ [samples]

Samples

Input #1

6 2 5
1 4
2 5

Output #1

5

![image](https://img.atcoder.jp/abc372/7a174918a45bdbdfb3457d9c62bea943.png)
The above figure represents the graph $G$. There are five ways for Takahashi to move:

*   Vertex $1 \to$ Vertex $2 \to$ Vertex $3 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6$
*   Vertex $1 \to$ Vertex $2 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $2$
*   Vertex $1 \to$ Vertex $2 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $4$
*   Vertex $1 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $2$
*   Vertex $1 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $4$

Input #2

10 0 200000

Output #2

Input #3

199 10 1326
122 39
142 49
164 119
197 127
188 145
69 80
6 120
24 160
18 154
185 27

Output #3

451022766

API Response (JSON)

{
  "problem": {
    "name": "Teleporting Takahashi 2",
    "description": {
      "content": "There is a simple directed graph $G$ with $N$ vertices and $N+M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $N+M$. Edge $i$ $(1 \\leq i \\leq N)$ goes from vertex $i$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc372_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a simple directed graph $G$ with $N$ vertices and $N+M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $N+M$.\nEdge $i$ $(1 \\leq i \\leq N)$ goes from vertex $i$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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