hdu-5816 Hearthstone(状压dp+概率期望)

题目链接：

Hearthstone

Time Limit: 2000/1000 MS (Java/Others)  

  Memory Limit: 65536/65536 K (Java/Others)

Problem Description

Hearthstone is an online collectible card game from Blizzard Entertainment. Strategies and luck are the most important factors in this game. When you suffer a desperate situation and your only hope depends on the top of the card deck, and you draw the only card to solve this dilemma. We call this "Shen Chou Gou" in Chinese.

Now you are asked to calculate the probability to become a "Shen Chou Gou" to kill your enemy in this turn. To simplify this problem, we assume that there are only two kinds of cards, and you don't need to consider the cost of the cards.
  -A-Card: If the card deck contains less than two cards, draw all the cards from the card deck; otherwise, draw two cards from the top of the card deck.
  -B-Card: Deal X damage to your enemy.

Note that different B-Cards may have different X values.
At the beginning, you have no cards in your hands. Your enemy has P Hit Points (HP). The card deck has N A-Cards and M B-Cards. The card deck has been shuffled randomly. At the beginning of your turn, you draw a card from the top of the card deck. You can use all the cards in your hands until you run out of it. Your task is to calculate the probability that you can win in this turn, i.e., can deal at least P damage to your enemy.

 

Input

The first line is the number of test cases T (T<=10). 
Then come three positive integers P (P<=1000), N and M (N+M<=20), representing the enemy’s HP, the number of A-Cards and the number of B-Cards in the card deck, respectively. Next line come M integers representing X (0<X<=1000) values for the B-Cards.

 

Output

For each test case, output the probability as a reduced fraction (i.e., the greatest common divisor of the numerator and denominator is 1). If the answer is zero (one), you should output 0/1 (1/1) instead.

 

Sample Input

2

3 1 2

1 2

3 5 10

1 1 1 1 1 1 1 1 1 1

 

Sample Output

1/3

46/273

 

题意：

 

给了n张A牌,m张B牌，A牌能取两张.B牌能打掉a[i]的血,问打掉至少p点血的概率是多少;

 

思路：

 

dp[i]表示i状态下的方案数,所求答案就是大于等于p点的状态的方案数总和与(n+m)!的比值;

状态的转移的话就是手中的A种牌在这个状态下是否还能拿牌,具体的可以看这个链接

 

AC代码：

 

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
//#include <bits/stdc++.h>
#include <stack>
#include <map>
 
using namespace std;
 
#define For(i,j,n) for(int i=j;i<=n;i++)
#define mst(ss,b) memset(ss,b,sizeof(ss));
 
typedef  long long LL;
 
template<class T> void read(T&num) {
    char CH; bool F=false;
    for(CH=getchar();CH<'0'||CH>'9';F= CH=='-',CH=getchar());
    for(num=0;CH>='0'&&CH<='9';num=num*10+CH-'0',CH=getchar());
    F && (num=-num);
}
int stk[70], tp;
template<class T> inline void print(T p) {
    if(!p) { puts("0"); return; }
    while(p) stk[++ tp] = p%10, p/=10;
    while(tp) putchar(stk[tp--] + '0');
    putchar('
');
}
 
const LL mod=1e9+7;
const double PI=acos(-1.0);
const int inf=1e9;
const int N=1e5+10;
const int maxn=(1<<20)+14;
const double eps=1e-12;

LL dp[maxn],f[23];
int num[maxn],p,n,m,a[30];
inline void Init()
{
    for(int i=0;i<maxn;i++)
    {
        for(int j=0;j<20;j++)
        {
            if(i&(1<<j))num[i]++;
        }
    }
    f[0]=1;
    for(int i=1;i<=20;i++)f[i]=f[i-1]*i;
}
int cal(int x)
{
    int sum=0;
    for(int i=0;i<m;i++)
    {
        if(x&(1<<i))sum+=a[i];
    }
    return sum;
}

LL gcd(LL x,LL y)
{
    if(y==0)return x;
    return gcd(y,x%y);
}
int counter(int x)
{
    int sum=0;
    for(int i=0;i<m;i++)
    {
        if(x&(1<<i))sum++;
    }
    return num[x]-2*sum+1;
}
int main()
{
    Init();
    int t;
    read(t);
    while(t--)
    {
        mst(dp,0);
        read(p);read(n);read(m);
        For(i,0,m-1)read(a[i]);
        LL ans=0;
        int tot=(1<<(n+m))-1;
        dp[0]=1;
        for(int i=0;i<=tot;i++)
        {
            if(!dp[i])continue;
            if(i==tot||counter(i)==0)
            {
                if(cal(i)>=p)ans=ans+dp[i]*f[n+m-num[i]];
            }
            else 
            {
                for(int j=0;j<n+m;j++)
                {
                    if(i&(1<<j))continue;
                    dp[i|(1<<j)]+=dp[i];
                }
            }
        }
        LL g=gcd(ans,f[n+m]);
        cout<<ans/g<<"/"<<f[n+m]/g<<"
";
    }
    return 0;
}