Score : $600$ points

问题陈述

AtCoder 镇有 $N$ 个交叉口和 $M$ 条道路。
第 $i$ 条道路连接交叉口 $A_i$ 和 $B_i$。

市长 Takahashi 决定在交叉口安装零个或多个监控摄像头。
每个交叉口可以安装零个或一个监控摄像头。

在 $2^N$ 种安装监控摄像头的方式中，有多少种方式能监控到偶数条道路？

这里，若满足以下条件，则认为第 $i$ 条道路受到了监控：

在交叉口 $A_i$ 和 $B_i$ 中的一个或两个位置安装了监控摄像头。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

AtCoder Town has $N$ intersections and $M$ roads.
Road $i$ connects Intersections $A_i$ and $B_i$.

Takahashi, the mayor, has decided to install zero or more surveillance cameras at the intersections.
Each intersection can be installed with zero or one surveillance camera.

How many of the $2^N$ ways to install surveillance cameras monitor an even number of roads?

Here, Road $i$ is said to be monitored when the following condition is satisfied:

a surveillance camera is installed at one or both of Intersections $A_i$ and $B_i$.

Constraints

• $2 \leq N \leq 40$
• $1 \leq M \leq \frac{N(N-1)}{2}$
• $1 \leq A_i \lt B_i \leq N$
• $(A_i,B_i) \neq (A_j,B_j)$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$

$A_2$ $B_2$

$\vdots$

$A_M$ $B_M$

Output

Print the answer.

Sample Input 1

3 2
1 2
2 3

Sample Output 1

6

The sets of towns to install surveillance cameras to satisfy the condition are: $ \{ \} , \{ 2 \} , \{ 1,2 \} ,\{1,3\},\{2,3\},\{1,2,3\}$.
Note that it is allowed to install no surveillance camera.

Sample Input 2

20 3
5 6
3 4
1 2

Sample Output 2

458752

The town may not be connected.

update @ 2024/3/10 09:44:22