Tri-Colored Paths

AtCoder

IDarc153_f

Time2000ms

Memory256MB

Difficulty—

You are given a simple connected undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the $i$\-th edge connects vertices $A_i$ and $B_i$. Find the number of ways to paint each edge of $G$ in color $1$, $2$, or $3$ so that the following condition is satisfied, modulo $998244353$. * There is a simple path in $G$ that contains an edge in color $1$, an edge in color $2$, and an edge in color $3$. What is a simple path? A simple path is a pair of a sequence of vertices $(v_1, \ldots, v_{k+1})$ and a sequence of edges $(e_1, \ldots, e_k)$ that satisfies the following: * $i\neq j \implies v_i\neq v_j$; * edge $e_i$ connects vertices $v_i$ and $v_{i+1}$. ## Constraints * $3 \leq N\leq 2\times 10^5$ * $3 \leq M\leq 2\times 10^5$ * $1 \leq A_i, B_i \leq N$ * The given graph is simple and connected. ## Input The input is given from Standard Input in the following format: $N$ $M$ $A_1$ $B_1$ $\vdots$ $A_M$ $B_M$ [samples]

Samples

Input #1

Output #1

0

Any simple path in $G$ contains two or fewer edges, so there is no way to satisfy the condition.

Input #2

Output #2

Input #3

Output #3

Input #4

Output #4

API Response (JSON)

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  "problem": {
    "name": "Tri-Colored Paths",
    "description": {
      "content": "You are given a simple connected undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the $i$\\-th edge connects vertices $A_i$ and $B_i$. Find the number of ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc153_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a simple connected undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$, and the $i$\\-th edge connects vertices $A_i$ and $B_i$.\nFind the number of ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments