Odd Degree

AtCoder

IDarc115_d

Time2000ms

Memory256MB

Difficulty—

Given is a simple undirected graph with $N$ vertices and $M$ edges, where the vertices are numbered $1, \ldots, N$ and the $i$\-th edge connects Vertex $A_i$ and Vertex $B_i$. For each $K(0 \leq K \leq N)$, find the number of spanning subgraphs (※) with exactly $K$ vertices with odd degrees. Since the answers can be enormous, print it modulo $998244353$. (※) A subgraph $H$ of $G$ is said to be a spanning subgraph of $G$ when the vertex set of $H$ equals the vertex set of $G$ and the edge set of $H$ is a subset of the edge set of $G$. ## Constraints * $1 \leq N \leq 5000$ * $0 \leq M \leq 5000$ * $1 \leq A_i , B_i \leq N$ * The given graph is simple, that is, it contains no self-loops and no multi-edges. ## Input Input is given from Standard Input in the following format: $N$ $M$ $A_1$ $B_1$ $A_2$ $B_2$ $:$ $A_M$ $B_M$ [samples]

Samples

Input #1

3 2
1 2
2 3

Output #1

1
0
3
0

Each spanning subgraph has the following number of vertices with odd degrees:

*   the subgraph with no edges has $0$ vertices with odd degrees;
*   the subgraph with just the edge connecting $1$ and $2$ has $2$ vertices with odd degrees;
*   the subgraph with just the edge connecting $2$ and $3$ has $2$ vertices with odd degrees;
*   the subgraph with both edges has $2$ vertices with odd degrees.

Input #2

4 2
1 2
3 4

Output #2

API Response (JSON)

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  "problem": {
    "name": "Odd Degree",
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      "content": "Given is a simple undirected graph with $N$ vertices and $M$ edges, where the vertices are numbered $1, \\ldots, N$ and the $i$\\-th edge connects Vertex $A_i$ and Vertex $B_i$. For each $K(0 \\leq K \\le",
      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
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    "is_sync": true,
    "sync_url": null,
    "sign": "arc115_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is a simple undirected graph with $N$ vertices and $M$ edges, where the vertices are numbered $1, \\ldots, N$ and the $i$\\-th edge connects Vertex $A_i$ and Vertex $B_i$. For each $K(0 \\leq K \\le...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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