Score : $500$ points

问题描述

你将得到一个包含 $N$ 个顶点和 $M$ 条边的简单连通无向图。（若图中没有重边且没有自环，则称该图为简单图。） 对于 $i = 1, 2, \ldots, M$，第 $i$ 条边连接顶点 $u_i$ 和顶点 $v_i$。

如果满足以下两个条件，那么序列 $(A_1, A_2, \ldots, A_k)$ 被称为长度为 $k$ 的路径：

• 对于所有 $i = 1, 2, \dots, k$，有 $1 \leq A_i \leq N$。
• 对于所有 $i = 1, 2, \ldots, k-1$，顶点 $A_i$ 和顶点 $A_{i+1}$ 直接由一条边相连。

空序列被视为长度为 $0$ 的路径。

令 $S = s_1s_2\ldots s_N$ 为长度为 $N$ 的字符串，仅由 $0$ 和 $1$ 构成。对于字符串 $S$，若路径 $A = (A_1, A_2, \ldots, A_k)$ 满足以下条件，则称其为相对于 $S$ 的好路径：

• 对于所有 $i = 1, 2, \ldots, N$，满足：
  • 如果 $s_i = 0$，则 $A$ 中含有偶数个 $i$。
  • 如果 $s_i = 1$，则 $A$ 中含有奇数个 $i$。

共有 $2^N$ 种可能的 $S$（换句话说，存在 $2^N$ 个由 $0$ 和 $1$ 组成、长度为 $N$ 的字符串）。找出所有这些 $S$ 中“最短好路径的长度”之和。

在本题的约束条件下，可以证明：对于任意由 $0$ 和 $1$ 组成的长度为 $N$ 的字符串 $S$，都至少存在一条相对于 $S$ 的好路径。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a simple connected undirected graph with $N$ vertices and $M$ edges. (A graph is said to be simple if it has no multi-edges and no self-loops.)
For $i = 1, 2, \ldots, M$, the $i$-th edge connects Vertex $u_i$ and Vertex $v_i$.

A sequence $(A_1, A_2, \ldots, A_k)$ is said to be a path of length $k$ if both of the following two conditions are satisfied:

• For all $i = 1, 2, \dots, k$, it holds that $1 \leq A_i \leq N$.
• For all $i = 1, 2, \ldots, k-1$, Vertex $A_i$ and Vertex $A_{i+1}$ are directly connected by an edge.

An empty sequence is regarded as a path of length $0$.

Let $S = s_1s_2\ldots s_N$ be a string of length $N$ consisting of $0$ and $1$. A path $A = (A_1, A_2, \ldots, A_k)$ is said to be a good path with respect to $S$ if the following conditions are satisfied:

• For all $i = 1, 2, \ldots, N$, it holds that:
  • if $s_i = 0$, then $A$ has even number of $i$'s.
  • if $s_i = 1$, then $A$ has odd number of $i$'s.

There are $2^N$ possible $S$ (in other words, there are $2^N$ strings of length $N$ consisting of $0$ and $1$). Find the sum of "the length of the shortest good path with respect to $S$" over all those $S$.

Under the Constraints of this problem, it can be proved that, for any string $S$ of length $N$ consisting of $0$ and $1$, there is at least one good path with respect to $S$.

Constraints

• $2 \leq N \leq 17$
• $N-1 \leq M \leq \frac{N(N-1)}{2}$
• $1 \leq u_i, v_i \leq N$
• The given graph is simple and connected.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_M$ $v_M$

Output

Print the answer.

Sample Input 1

3 2
1 2
2 3

Sample Output 1

14

• For $S = 000$, the empty sequence $()$ is the shortest good path with respect to $S$, whose length is $0$.
• For $S = 100$, $(1)$ is the shortest good path with respect to $S$, whose length is $1$.
• For $S = 010$, $(2)$ is the shortest good path with respect to $S$, whose length is $1$.
• For $S = 110$, $(1, 2)$ is the shortest good path with respect to $S$, whose length is $2$.
• For $S = 001$, $(3)$ is the shortest good path with respect to $S$, whose length is $1$.
• For $S = 101$, $(1, 2, 3, 2)$ is the shortest good path with respect to $S$, whose length is $4$.
• For $S = 011$, $(2, 3)$ is the shortest good path with respect to $S$, whose length is $2$.
• For $S = 111$, $(1, 2, 3)$ is the shortest good path with respect to $S$, whose length is $3$.

Therefore, the sought answer is $0 + 1 + 1 + 2 + 1 + 4 + 2 + 3 = 14$.

Sample Input 2

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

Sample Output 2

108

update @ 2024/3/10 10:30:09