102136: [AtCoder]ABC213 G - Connectivity 2

Problem Statement

Given is a simple undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1,2,\dots,N$, the edges are numbered $1,2,\dots,M$, and Edge $i$ connects Vertex $a_i$ and Vertex $b_i$.
Consider removing zero or more edges from $G$ to get a new graph $H$. There are $2^M$ graphs that we can get as $H$. Among them, find the number of such graphs that Vertex $1$ and Vertex $k$ are directly or indirectly connected, for each integer $k$ such that $2 \leq k \leq N$.
Since the counts may be enormous, print them modulo $998244353$.

Constraints

• $2 \leq N \leq 17$
• $0 \leq M \leq \frac{N(N-1)}{2}$
• $1 \leq a_i \lt b_i \leq N$
• $(a_i, b_i) \neq (a_j, b_j)$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$a_1$ $b_1$
$\vdots$
$a_M$ $b_M$

Output

Print $N-1$ lines. The $i$-th line should contain the answer for $k = i + 1$.

Sample Input 1

3 2
1 2
2 3

Sample Output 1

2
1

We can get the following graphs as $H$.

• The graph with no edges. Vertex $1$ is disconnected from any other vertex.
• The graph with only the edge connecting Vertex $1$ and $2$. Vertex $2$ is reachable from Vertex $1$.
• The graph with only the edge connecting Vertex $2$ and $3$. Vertex $1$ is disconnected from any other vertex.
• The graph with both edges. Vertex $2$ and $3$ are reachable from Vertex $1$.

Sample Input 2

5 6
1 2
1 4
1 5
2 3
2 5
3 4

Sample Output 2

43
31
37
41

Sample Input 3

2 0

Sample Output 3

0

问题描述

给定一个具有$N$个顶点和$M$条边的简单无向图$G$。顶点编号为$1,2,\dots,N$，边编号为$1,2,\dots,M$，边$i$连接顶点$a_i$和顶点$b_i$。
考虑从$G$中删除零条或多条边，得到一个新的图$H$。我们可以通过$H$得到$2^M$个图。其中，对于每个满足$2 \leq k \leq N$的整数$k$，找出顶点$1$和顶点$k$直接或间接连接的图的数量。
由于计数可能非常庞大，请将它们对$998244353$取模。

约束

• $2 \leq N \leq 17$
• $0 \leq M \leq \frac{N(N-1)}{2}$
• $1 \leq a_i \lt b_i \leq N$
• 如果$i \neq j$，则$(a_i, b_i) \neq (a_j, b_j)$。
• 输入中的所有值都是整数。

输入

输入将从标准输入按以下格式给出：

$N$ $M$
$a_1$ $b_1$
$\vdots$
$a_M$ $b_M$

输出

打印$N-1$行。第$i$行应包含对于$k = i + 1$的答案。

样例输入 1

3 2
1 2
2 3

样例输出 1

2
1

我们可以得到以下作为$H$的图。

• 没有边的图。顶点$1$与任何其他顶点断开连接。
• 仅包含连接顶点$1$和$2$的边的图。顶点$2$可以从顶点$1$到达。
• 仅包含连接顶点$2$和$3$的边的图。顶点$1$与任何其他顶点断开连接。
• 包含两条边的图。顶点$2$和$3$可以从顶点$1$到达。

样例输入 2

5 6
1 2
1 4
1 5
2 3
2 5
3 4

样例输出 2

43
31
37
41

样例输入 3

2 0

样例输出 3

0