Endless Walk

AtCoder

IDabc245_f

Time2000ms

Memory256MB

Difficulty—

We have a simple directed graph $G$ with $N$ vertices and $M$ edges. The vertices are labeled as Vertex $1$, Vertex $2$, $\ldots$, Vertex $N$. The $i$\-th edge $(1\leq i\leq M)$ goes from Vertex $U_i$ to Vertex $V_i$. Takahashi will start at a vertex and repeatedly travel on $G$ from one vertex to another along a directed edge. How many vertices of $G$ have the following condition: Takahashi can start at that vertex and continue traveling indefinitely by carefully choosing the path? ## Constraints * $1 \leq N \leq 2\times 10^5$ * $0 \leq M \leq \min(N(N-1), 2\times 10^5)$ * $1 \leq U_i,V_i\leq N$ * $U_i\neq V_i$ * $(U_i,V_i)\neq (U_j,V_j)$ if $i\neq j$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $U_1$ $V_1$ $U_2$ $V_2$ $\vdots$ $U_M$ $V_M$ [samples]

Samples

Input #1

Output #1

4

When starting at Vertex $2$, Takahashi can continue traveling indefinitely: $2$ $\to$ $3$ $\to$ $4$ $\to$ $2$ $\to$ $3$ $\to$ $\cdots$ The same goes when starting at Vertex $3$ or Vertex $4$. From Vertex $1$, he can first go to Vertex $2$ and then continue traveling indefinitely again.  
On the other hand, from Vertex $5$, he cannot move at all.
Thus, four vertices ―Vertex $1$, $2$, $3$, and $4$― satisfy the conditions, so $4$ should be printed.

Input #2

3 2
1 2
2 1

Output #2

2

Note that, in a simple directed graph, there may be two edges in opposite directions between the same pair of vertices. Additionally, $G$ may not be connected.

API Response (JSON)

{
  "problem": {
    "name": "Endless Walk",
    "description": {
      "content": "We have a simple directed graph $G$ with $N$ vertices and $M$ edges. The vertices are labeled as Vertex $1$, Vertex $2$, $\\ldots$, Vertex $N$. The $i$\\-th edge $(1\\leq i\\leq M)$ goes from Vertex $U_i$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc245_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a simple directed graph $G$ with $N$ vertices and $M$ edges. The vertices are labeled as Vertex $1$, Vertex $2$, $\\ldots$, Vertex $N$. The $i$\\-th edge $(1\\leq i\\leq M)$ goes from Vertex $U_i$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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