Unique Path

AtCoder

IDagc038_d

Time2000ms

Memory256MB

Difficulty—

Snuke's mother gave Snuke an undirected graph consisting of $N$ vertices numbered $0$ to $N-1$ and $M$ edges. This graph was connected and contained no parallel edges or self-loops. One day, Snuke broke this graph. Fortunately, he remembered $Q$ clues about the graph. The $i$\-th clue ($0 \leq i \leq Q-1$) is represented as integers $A_i,B_i,C_i$ and means the following: * If $C_i=0$: there was exactly one simple path (a path that never visits the same vertex twice) from Vertex $A_i$ to $B_i$. * If $C_i=1$: there were two or more simple paths from Vertex $A_i$ to $B_i$. Snuke is not sure if his memory is correct, and worried whether there is a graph that matches these $Q$ clues. Determine if there exists a graph that matches Snuke's memory. ## Constraints * $2 \leq N \leq 10^5$ * $N-1 \leq M \leq N \times (N-1)/2$ * $1 \leq Q \leq 10^5$ * $0 \leq A_i,B_i \leq N-1$ * $A_i \neq B_i$ * $0 \leq C_i \leq 1$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $Q$ $A_0$ $B_0$ $C_0$ $A_1$ $B_1$ $C_1$ $\vdots$ $A_{Q-1}$ $B_{Q-1}$ $C_{Q-1}$ [samples]

Samples

Input #1

Output #1

Yes

For example, consider a graph with edges $(0,1),(1,2),(1,4),(2,3),(2,4)$. This graph matches the clues.

Input #2

Output #2

No

Input #3

Output #3

No

API Response (JSON)

{
  "problem": {
    "name": "Unique Path",
    "description": {
      "content": "Snuke's mother gave Snuke an undirected graph consisting of $N$ vertices numbered $0$ to $N-1$ and $M$ edges. This graph was connected and contained no parallel edges or self-loops. One day, Snuke bro",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc038_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke's mother gave Snuke an undirected graph consisting of $N$ vertices numbered $0$ to $N-1$ and $M$ edges. This graph was connected and contained no parallel edges or self-loops.\nOne day, Snuke bro...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments