【2019牛客暑期多校训练营（第五场）- E】independent set 1（最大独立集，状压dp）

题干：

链接：https://ac.nowcoder.com/acm/contest/885/E
来源：牛客网

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′⊆VV' \subseteq VV′⊆V

* edge (a,b)∈E′(a,b) \in E'(a,b)∈E′ if and only if a∈V′,b∈V′a \in V', b \in V'a∈V′,b∈V′, and edge (a,b)∈E(a, b) \in E(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 2n2^n2n possible induced subgraphs of the given graph. Please calculate the sum of the 2n2^n2n such cardinalities.

输入描述:

The first line contains two integers n and m (2≤n≤26,0≤m≤n×(n−1)22 \le n \le 26, 0 \le m \le \frac{n \times (n-1)}{2}2≤n≤26,0≤m≤2n×(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,yix_i, y_ixi,yi (0≤xi<yi<n0 \le x_i < y_i < n0≤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.

示例1

输入

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3 2
0 1
0 2

输出

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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.

示例2

输入

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输出

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题目大意：

给一个n个点m条边的无向图，求所有子图（包括本身）的最大独立集元素个数之和。（2<=n<=26）

解题报告：

直接状压dp，求出对于每一个子图的最大独立集的元素之和就可以了。

转移方程：dp[sta]=max(dp[最大独立集不包含这个点],dp[包含这个点])。

然后卡空间，所以要char类型。

AC代码：

#include<cstdio>
#include<iostream>
#include<algorithm>
#include<queue>
#include<map>
#include<vector>
#include<set>
#include<string>
#include<cmath>
#include<cstring>
#define FF first
#define SS second
#define ll long long
#define pb push_back
#define pm make_pair
using namespace std;
typedef pair<int,int> PII;
const int MAX = 1<<26;
int b[MAX];
char p[MAX],dp[MAX];
int n,m;
int main()
{cin>>n>>m;int ans = 0;for(int u,v,i = 1; i<=m; i++) {scanf("%d%d",&u,&v);b[u] |= 1<<v;b[v] |= 1<<u;                }dp[0] = 0;for(int sta = 1; sta < (1<<n); sta++) {int x = __builtin_ffs(sta);x--;dp[sta] = max((int)dp[sta-(1<<x)],dp[sta - (1<<x) - (sta & b[x])] + 1);ans += dp[sta];}printf("%d\n",ans);return 0 ;
}