Codeforces Round #363 (Div. 2)E. LRU

E. LRU

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

While creating high loaded systems one should pay a special attention to caching. This problem will be about one of the most popular caching algorithms called LRU (Least Recently Used).

Suppose the cache may store no more than k objects. At the beginning of the workflow the cache is empty. When some object is queried we check if it is present in the cache and move it here if it's not. If there are more than k objects in the cache after this, the least recently used one should be removed. In other words, we remove the object that has the smallest time of the last query.

Consider there are n videos being stored on the server, all of the same size. Cache can store no more than k videos and caching algorithm described above is applied. We know that any time a user enters the server he pick the video i with probability pi. The choice of the video is independent to any events before.

The goal of this problem is to count for each of the videos the probability it will be present in the cache after 10100 queries.

Input

The first line of the input contains two integers n and k (1 ≤ k ≤ n ≤ 20) — the number of videos and the size of the cache respectively. Next line contains n real numbers pi (0 ≤ pi ≤ 1), each of them is given with no more than two digits after decimal point.

It's guaranteed that the sum of all pi is equal to 1.

Output

Print n real numbers, the i-th of them should be equal to the probability that the i-th video will be present in the cache after 10100queries. You answer will be considered correct if its absolute or relative error does not exceed 10 - 6.

Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .

Examples

input

3 1
0.3 0.2 0.5

output

0.3 0.2 0.5 

input

2 1
0.0 1.0

output

0.0 1.0 

input

3 2
0.3 0.2 0.5

output

0.675 0.4857142857142857 0.8392857142857143 

input

3 3
0.2 0.3 0.5

output

1.0 1.0 1.0 

/* ***********************************************
Author        :guanjun
Created Time  :2016/7/26 10:32:35
File Name     :1.cpp
************************************************ */
#include <iostream>
#include <cstring>
#include <cstdlib>
#include <stdio.h>
#include <algorithm>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <string>
#include <math.h>
#include <stdlib.h>
#include <iomanip>
#include <list>
#include <deque>
#include <stack>
#define ull unsigned long long
#define ll long long
#define mod 90001
#define INF 0x3f3f3f3f
#define maxn 10010
#define cle(a) memset(a,0,sizeof(a))
const ull inf = 1LL << 61;
const double eps=1e-5;
using namespace std;
priority_queue<int,vector<int>,greater<int> >pq;
struct Node{
    int x,y;
};
struct cmp{
    bool operator()(Node a,Node b){
        if(a.x==b.x) return a.y> b.y;
        return a.x>b.x;
    }
};

bool cmp(int a,int b){
    return a>b;
}
int n,k;
double p[30];
double dp[1<<20];
double ans[30];

int main()
{
    #ifndef ONLINE_JUDGE
    freopen("in.txt","r",stdin);
    #endif
    //freopen("out.txt","w",stdout);
    cin>>n>>k;
    dp[0]=1;
    cle(ans);
    for(int i=0;i<n;i++)cin>>p[i];
    for(int i=0;i<(1<<n);i++)if(dp[i]){
        double s=1;
        for(int j=0;j<n;j++)
            if(i&(1<<j))s-=p[j];
        if(s<1e-9||__builtin_popcount(i)==k){
            for(int j=0;j<n;j++){
                if(i&(1<<j))ans[j]+=dp[i];
            }
        }
        else{
            for(int j=0;j<n;j++)
                if((i&(1<<j))==0)dp[i|(1<<j)]+=dp[i]*p[j]/s;
        }
    }
    for(int i=0;i<n;i++){
        printf("%.7f ", ans[i]);
    }
    return 0;
}

dp[1110]怎么由dp[1100]转移过去呢

dp[1110]=dp[1100]*p[3]/(1-p[1]-p[2])

p[3]/(1-p[1]-p[2])是一个整体  代表在剩下的没选的视频中选择第3个视频的概率