independent set 1

 independent set 1

时间限制：C/C++ 1秒，其他语言2秒
空间限制：C/C++ 102400K，其他语言204800K
64bit IO Format: %lld

题目描述

Note: For C++ languages, the memory limit is 100 MB. For other languages, the memory limit is 200 MB.

In graph theory, an independent set is a set of nonadjacent vertices in a graph. Moreover, an independent set is maximum if it has the maximum cardinality (in this case, the number of vertices) across all independent sets in a graph.

An induced subgraph G'(V', E') of a graph G(V, E) is a graph that satisfies:

* V′⊆V

* edge (a,b)∈E′ if and only if a∈V′,b∈V′, and edge (a,b)∈E;


Now, given an undirected unweighted graph consisting of n vertices and m edges. This problem is about the cardinality of the maximum independent set of each of the 2^n possible induced subgraphs of the given graph. Please calculate the sum of the 2^n such cardinalities.

输入描述:

The first line contains two integers n and m (2≤n≤26,0≤m≤n×(n−1)​) --- the number of vertices and the number of edges, respectively. Next m lines describe edges: the i-th line contains two integers xi,yi (0≤xi<yi<n) --- the indices (numbered from 0 to n - 1) of vertices connected by the i-th edge.

The graph does not have any self-loops or multiple edges.

输出描述:

Print one line, containing one integer represents the answer.

输入

3 2
0 1
0 2

输出

9

说明

The cardinalities of the maximum independent set of every subset of vertices are: {}: 0, {0}: 1, {1}: 1, {2}: 1, {0, 1}: 1, {0, 2}: 1, {1, 2}: 2, {0, 1, 2}: 2. So the sum of them are 9.
链接：https://ac.nowcoder.com/acm/contest/885/E
来源：牛客网

---

题意：求一个图的2^n种子图的最大点独立集。
思路：

 •我们可以用一个 n-bit 2 进制整数来表示一个点集，第 i 个 bit 是 1 就代表点集包含第 i 个 点，若是 0 则不包含 
 • 每个点相邻的点也可以用一个 n-bit 2 进制整数表示，计做 ci，若第 i 个点和第 j 个点相邻， ci 的第 j 个 bit 是 1，否则是 0
 • 记 x 的最低位且是 1 的 bit 的位置是 lbx
 • 令 dp[x] 代表点集 x 的最大独立集 size，那么我们能够根据点 lbx 是否属于最大独立集来列 出以下关系式：
 dp[x] = max(dp[x - (1<<lbx)], dp[x & (~clb_x)] + 1) (使用 c 语言运算符)

 •高效位运算参考博客：https://blog.csdn.net/yuer158462008/article/details/46383635

#include<bits/stdc++.h>
using namespace std;
char dp[1<<26];
int Map[26]={0};
int max(char a,int b)
{
    if(a>b)return a;
    return b;
}
int main()
{    
    int n,m;
    scanf("%d %d",&n,&m);
    
    while(m--)
    {
        int u,v;
        scanf("%d %d",&u,&v);
        Map[u]|=(1<<v);
        Map[v]|=(1<<u);
    }
    for(int i=0;i<n;i++)Map[i]|=1<<i;
    
    int upper=1<<n;
    long long ans=0;
    for(int i=1;i<upper;i++)
    {
        int lbx=__builtin_ctz(i);
        dp[i] = max(dp[i - (1<<lbx)] , dp[i & (~Map[lbx])] + 1);
        ans+=dp[i];
    }
    printf("%lld
",ans);
    return 0;
}