102445: [AtCoder]ABC244 F - Shortest Good Path
Description
Score : $500$ points
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
Input
Output
问题描述
给你一个有$N$个顶点和$M$条边的简单连通无向图（一个图被称为简单，如果它没有多边和自环）。
对于$i=1,2,\ldots,M$，第$i$条边连接顶点$u_i$和顶点$v_i$。
一个序列$(A_1, A_2, \ldots, A_k)$被称为长度为$k$的路径，如果满足以下两个条件：
- 对于所有$i=1,2,\dots,k$，都有$1 \leq A_i \leq N$。
- 对于所有$i=1,2,\ldots,k-1$，顶点$A_i$和顶点$A_{i+1}$之间由一条边直接连接。
空序列被视为长度为$0$的路径。
让$S = s_1s_2\ldots s_N$是一个由$0$和$1$组成的长度为$N$的字符串。一个路径$A = (A_1, A_2, \ldots, A_k)$被称为关于$S$的好路径，如果满足以下条件：
- 对于所有$i=1,2,\ldots,N$，都有：
- 如果$s_i = 0$，那么$A$有偶数个$i$。
- 如果$s_i = 1$，那么$A$有奇数个$i$。
有$2^N$种可能的$S$（换句话说，有$2^N$个长度为$N$的由$0$和$1$组成的字符串）。对于所有这些$S$，求“关于$S$的最短好路径的长度”的和。
在本问题的约束下，可以证明，对于任何长度为$N$的由$0$和$1$组成的字符串$S$，都至少有一个关于$S$的好路径。
约束条件
- $2 \leq N \leq 17$
- $N-1 \leq M \leq \frac{N(N-1)}{2}$
- $1 \leq u_i, v_i \leq N$
- 给定的图是简单且连通的。
- 输入中的所有值都是整数。
输入
输入标准输入，格式如下：
$N$ $M$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$
输出
打印答案。
样例输入 1
3 2 1 2 2 3
样例输出 1
14
- 对于$S = 000$，空序列$()$是关于$S$的最短好路径，其长度为$0$。
- 对于$S = 100$，$(1)$是关于$S$的最短好路径，其长度为$1$。
- 对于$S = 010$，$(2)$是关于$S$的最短好路径，其长度为$1$。
- 对于$S = 110$，$(1, 2)$是关于$S$的最短好路径，其长度为$2$。
- 对于$S = 001$，$(3)$是关于$S$的最短好路径，其长度为$1$。
- 对于$S = 101$，$(1, 2, 3, 2)$是关于$S$的最短好路径，其长度为$4$。
- 对于$S = 011$，$(2, 3)$是关于$S$的最短好路径，其长度为$2$。
- 对于$S = 111$，$(1, 2, 3)$是关于$S$的最短好路径，其长度为$3$。
因此，所求答案为$0 + 1 + 1 + 2 + 1 + 4 + 2 + 3 = 14$。
样例输入 2
5 5 4 2 2 3 1 3 2 1 1 5
样例输出 2
108