Shortest Good Path

题目描述

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 $S$.

输入格式

Input is given from Standard Input in the following format: $N$ $M$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$

输出格式

Print the answer.

3 2
1 2
2 3

14

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

108

提示

$2 \leq N \leq 17$ $N-1 \leq M \leq \frac{N(N-1)}{2}$ $1 \leq u_i, v_i \leq N$ The given graph is simple and connected. All values in input are integers.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 $0 + 1 + 1 + 2 + 1 + 4 + 2 + 3 = 14$.

标签: AtCoder|abc244F

来源

AtCoder|abc244F