URAL 1776 C - Anniversary Firework DP

C - Anniversary Firework
Time Limit: 20 Sec

Memory Limit: 256 MB

题目连接

http://acm.hust.edu.cn/vjudge/contest/view.action?cid=87643#problem/B

Description

Denis has to prepare the Ural State University 90th anniversary firework. He bought n rockets and started to think of the way he should launch them. After a pair of sleepless nights he invented the following algorithm.

All n rockets are placed on the surface in a single line. The interval between two consecutive salvos is ten seconds. The leftmost and the rightmost rocket are launched in the first salvo. After i salvos are fired, all non-empty segments between two neighboring launched rockets are considered. One rocket is chosen randomly and uniformly at each of these segments. All chosen rockets are launched in the (i + 1)-st salvo. Algorithm runs until all rockets are launched.

Calculate the average duration in seconds of such a firework.

Input

The only input line contains an integer n (3 ≤ n ≤ 400) , which is the number of rockets bought by Denis.

Output

Output the expected duration of the firework in seconds, with absolute or relative error not exceeding 10 ⁻⁶.

Sample Input

Sample Output

26.66666666666

HINT

题意

有n个火箭，每次你都会在已经放了火箭的区间内随机选择一个放火箭

然后问你把火箭全部放完的期望时间是多少

题解：

首先，期望取最大值这个是错误的

这个dp[i][j]应该表示长度为i的，放j个火箭的概率是多少

直接dfs解决就好了

代码：

//qscqesze

#include <cstdio>

#include <cmath>

#include <cstring>

#include <ctime>

#include <iostream>

#include <algorithm>

#include <set>

#include <vector>

#include <sstream>

#include <queue>

#include <typeinfo>

#include <fstream>

#include <map>

#include <stack>

typedef long long ll;

using namespace std;

//freopen("D.in","r",stdin);

//freopen("D.out","w",stdout);

#define sspeed ios_base::sync_with_stdio(0);cin.tie(0)

#define maxn 5000

#define mod 10007

#define eps 1e-9

int Num;

//const int inf=0x7fffffff;   //нчоч╢С

const int inf=0x3f3f3f3f;

inline ll read()

{

    ll x=,f=;char ch=getchar();

    while(ch<''||ch>''){if(ch=='-')f=-;ch=getchar();}

    while(ch>=''&&ch<=''){x=x*+ch-'';ch=getchar();}

    return x*f;

}

//**************************************************************************************

int v[][];

double dp[][];

double dfs(int x,int y)

{

    if(x==)return 1.0;

    if(y==)return 0.0;

    if(v[x][y])

        return dp[x][y];

    double p = 1.0/(x*1.0);

    double ans=;

    for(int i=;i<=x;i++)

        ans+=p*(dfs(i-,y-)*dfs(x-i,y-));

    v[x][y]=;

    dp[x][y]=ans;

    return ans;

}

int main()

{

    int n;

    cin>>n;

    double ans=;

    for(int i=;i<=n-;i++)

        ans+=(dfs(n-,i)-dfs(n-,i-))*(1.0**i);

    printf("%.10lf\n",ans);

}