[Codeforces443D]Andrey and Problem

2014-06-26
竞赛题解 / 贪心
Codeforces

描述 Description

Andrey needs one more problem to conduct a programming contest. He has n friends who are always willing to help. He can ask some of them to come up with a contest problem. Andrey knows one value for each of his fiends — the probability that this friend will come up with a problem if Andrey asks him.

Help Andrey choose people to ask. As he needs only one problem, Andrey is going to be really upset if no one comes up with a problem or if he gets more than one problem from his friends. You need to choose such a set of people that maximizes the chances of Andrey not getting upset.

输入格式 InputFormat

The first line contains a single integer n (1 ≤ n ≤ 100) — the number of Andrey’s friends. The second line contains n real numbers pi (0.0 ≤ pi ≤ 1.0) — the probability that the i-th friend can come up with a problem. The probabilities are given with at most 6 digits after decimal point.

输出格式 OutputFormat

Print a single real number — the probability that Andrey won’t get upset at the optimal choice of friends. The answer will be considered valid if it differs from the correct one by at most 10-9.

样例输入 SampleInput

4

0.1 0.2 0.3 0.8

​
## 样例输出 SampleOutput >0.800000000000

数据范围和注释 Hint

In the first sample the best strategy for Andrey is to ask only one of his friends, the most reliable one.

In the second sample the best strategy for Andrey is to ask all of his friends to come up with a problem. Then the probability that he will get exactly one problem is 0.1·0.8 + 0.9·0.2 = 0.26.

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Codeforces 443D

代码 Code

这样做法是可行的，但是证明的话非常麻烦。。。

#include <stdio.h>
#include <iostream>
#include <cstring>
#include <algorithm>
#include <cmath>
using namespace std;
#define inf 0x7fffffff/27.11
double a[1001],f[1001][2];
int i,j,t,n,m,l,r,k,z,y,x;
double ans;
inline bool comp(double a,double b)
{
    return a>b;
}
int main()
{
    scanf("%d",&n);
    for (i=1;i<=n;i++) scanf("%lf",&a[i]);
    sort(a+1,a+n+1,comp);
    f[0][0]=1;
    for (i=1;i<=n;i++)
    {
        f[i][0]=f[i-1][0]*(1-a[i]);
        f[i][1]=f[i-1][0]*a[i]+f[i-1][1]*(1-a[i]);
    }
    ans=-inf;
    for (i=1;i<=n;i++) ans=max(ans,f[i][1]);
    printf("%.12lf",ans);
}