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HDU-4336 Card Collector 概率DP

　　题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=4336

　　题意：买食品收集n个卡片，每个卡片的概率分别是pi，且Σp[i]<=1，求收集n个卡片需要买的食品数的期望。

　　压缩DP：把每个食品用二进制表示，0和1分别表示没有卡片和已经收集到此卡片的期望，则

　　　　 f[s]=(1-Σp[i])*f[s]+Σp[j]*f[s]+Σp[k]*f[s|(1<<k)]

　　　　　　s表示状态，i表示所有卡片编号，j表示s状态中已经有的卡片编号，k表示s状态中没有的卡片编号

　　-> Σp[i]*f[s]=Σp[i]*f[s|(1<<i)]

　　或者容斥原理做：

　　压缩DP:

 1 //STATUS:C++_AC_281MS_7128KB
 2 #include <functional>
 3 #include <algorithm>
 4 #include <iostream>
 5 //#include <ext/rope>
 6 #include <fstream>
 7 #include <sstream>
 8 #include <iomanip>
 9 #include <numeric>
10 #include <cstring>
11 #include <cassert>
12 #include <cstdio>
13 #include <string>
14 #include <vector>
15 #include <bitset>
16 #include <queue>
17 #include <stack>
18 #include <cmath>
19 #include <ctime>
20 #include <list>
21 #include <set>
22 #include <map>
23 using namespace std;
24 //#pragma comment(linker,"/STACK:102400000,102400000")
25 //using namespace __gnu_cxx;
26 //define
27 #define pii pair<int,int>
28 #define mem(a,b) memset(a,b,sizeof(a))
29 #define lson l,mid,rt<<1
30 #define rson mid+1,r,rt<<1|1
31 #define PI acos(-1.0)
32 //typedef
33 typedef __int64 LL;
34 typedef unsigned __int64 ULL;
35 //const
36 const int N=(1<<20)+10;
37 const int INF=0x3f3f3f3f;
38 const int MOD= 1000000007,STA=8000010;
39 const LL LNF=1LL<<55;
40 const double EPS=1e-9;
41 const double OO=1e30;
42 const int dx[4]={-1,0,1,0};
43 const int dy[4]={0,1,0,-1};
44 const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
45 //Daily Use ...
46 inline int sign(double x){return (x>EPS)-(x<-EPS);}
47 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;}
48 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}
49 template<class T> inline T lcm(T a,T b,T d){return a/d*b;}
50 template<class T> inline T Min(T a,T b){return a<b?a:b;}
51 template<class T> inline T Max(T a,T b){return a>b?a:b;}
52 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);}
53 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);}
54 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));}
55 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));}
56 //End
57 
58 double p[23],f[N];
59 int n;
60 
61 int main(){
62  //   freopen("in.txt","r",stdin);
63     int i,j,up;
64     double s;
65     while(~scanf("%d",&n))
66     {
67         for(i=0;i<n;i++){
68             scanf("%lf",&p[i]);
69         }
70         up=(1<<n)-1;
71         f[up]=0;
72         for(i=up-1;i>=0;i--){
73             f[i]=1;s=0;
74             for(j=0;j<n;j++){
75                 if(i&(1<<j))continue;
76                 f[i]+=p[j]*f[i|(1<<j)];
77                 s+=p[j];
78             }
79             f[i]/=s;
80         }
81 
82         printf("%lf
",f[0]);
83     }
84     return 0;
85 }

　　容斥原理：

 1 //STATUS:C++_AC_203MS_244KB
 2 #include <functional>
 3 #include <algorithm>
 4 #include <iostream>
 5 //#include <ext/rope>
 6 #include <fstream>
 7 #include <sstream>
 8 #include <iomanip>
 9 #include <numeric>
10 #include <cstring>
11 #include <cassert>
12 #include <cstdio>
13 #include <string>
14 #include <vector>
15 #include <bitset>
16 #include <queue>
17 #include <stack>
18 #include <cmath>
19 #include <ctime>
20 #include <list>
21 #include <set>
22 #include <map>
23 using namespace std;
24 //#pragma comment(linker,"/STACK:102400000,102400000")
25 //using namespace __gnu_cxx;
26 //define
27 #define pii pair<int,int>
28 #define mem(a,b) memset(a,b,sizeof(a))
29 #define lson l,mid,rt<<1
30 #define rson mid+1,r,rt<<1|1
31 #define PI acos(-1.0)
32 //typedef
33 typedef __int64 LL;
34 typedef unsigned __int64 ULL;
35 //const
36 const int N=(1<<20)+10;
37 const int INF=0x3f3f3f3f;
38 const int MOD= 1000000007,STA=8000010;
39 const LL LNF=1LL<<55;
40 const double EPS=1e-9;
41 const double OO=1e30;
42 const int dx[4]={-1,0,1,0};
43 const int dy[4]={0,1,0,-1};
44 const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};
45 //Daily Use ...
46 inline int sign(double x){return (x>EPS)-(x<-EPS);}
47 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;}
48 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}
49 template<class T> inline T lcm(T a,T b,T d){return a/d*b;}
50 template<class T> inline T Min(T a,T b){return a<b?a:b;}
51 template<class T> inline T Max(T a,T b){return a>b?a:b;}
52 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);}
53 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);}
54 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));}
55 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));}
56 //End
57 
58 double p[23];
59 int n;
60 
61 int main(){
62  //   freopen("in.txt","r",stdin);
63     int i,j,up,cnt;
64     double ans,s;
65     while(~scanf("%d",&n))
66     {
67         for(i=0;i<n;i++){
68             scanf("%lf",&p[i]);
69         }
70         up=(1<<n)-1;ans=0;
71         for(i=1;i<=up;i++){
72             s=0;
73             for(j=cnt=0;j<n;j++){
74                 if(i&(1<<j)){
75                     cnt++;
76                     s+=p[j];
77                 }
78             }
79             if(cnt&1)ans+=1/s;
80             else ans-=1/s;
81         }
82 
83         printf("%lf
",ans);
84     }
85     return 0;
86 }

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原文地址：https://www.cnblogs.com/zhsl/p/3244041.html