ZOJ-3640 Help Me Escape （概率DP）

Help Me Escape

http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemId=4808

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Time Limit: 2 Seconds       Memory Limit: 32768 KB

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Background

    If thou doest well, shalt thou not be accepted? and if thou doest not well, sin lieth at the door. And unto thee shall be his desire, and thou shalt rule over him. 
    And Cain talked with Abel his brother: and it came to pass, when they were in the field, that Cain rose up against Abel his brother, and slew him. 
    And the LORD said unto Cain, Where is Abel thy brother? And he said, I know not: Am I my brother's keeper? 
    And he said, What hast thou done? the voice of thy brother's blood crieth unto me from the ground. 
    And now art thou cursed from the earth, which hath opened her mouth to receive thy brother's blood from thy hand; 
    When thou tillest the ground, it shall not henceforth yield unto thee her strength; a fugitive and a vagabond shalt thou be in the earth.

—— Bible Chapter 4

Now Cain is unexpectedly trapped in a cave with N paths. Due to LORD's punishment, all the paths are zigzag and dangerous. The difficulty of the ith path is ci.

Then we define f as the fighting capacity of Cain. Every day, Cain will be sent to one of the N paths randomly.

Suppose Cain is in front of the ith path. He can successfully take ti days to escape from the cave as long as his fighting capacity f is larger than ci. Otherwise, he has to keep trying day after day. However, if Cain failed to escape, his fighting capacity would increase ci as the result of actual combat. (A kindly reminder: Cain will never died.)

As for ti, we can easily draw a conclusion that ti is closely related to ci. Let's use the following function to describe their relationship:

After D days, Cain finally escapes from the cave. Please output the expectation of D.

Input

The input consists of several cases. In each case, two positive integers N and f (n ≤ 100, f ≤ 10000) are given in the first line. The second line includes N positive integers ci (ci ≤ 10000, 1 ≤ i ≤ N)

Output

For each case, you should output the expectation(3 digits after the decimal point).

Sample Input

3 1
1 2 3

Sample Output

6.889

题目大意：该隐初始有战斗力f，每天会随机传送值n点中的一点，第i点的有一个值为c[i]，若f>c[i]，则该隐会花t[i]（t[i]=floor((1+sqrt(5))/2*c[i]*c[i])）天逃离，否则f增加c[i]，求逃离时经过的天数的期望？

状态没设对，导致直接没法转移

应该设dp[f]表示战斗力为f时，逃离时经过的天数的期望

则：

①f>c[i]时，本次能从i点逃离，dp[f]=t[i]/n;

②f<=c[i]时，本次不能从i点逃离，dp[f]=∑(dp[f+c[i]]+1)/n;

由于各状态是离散的，所以只能通过dfs进行状态转移

#include 
#include 
#include 

using namespace std;

const int MAXN=105;
const double TMP=0.5*(1.0+sqrt(5.0));
const double EPS=0.000001;

int n,f;
double dp[20005],c[MAXN];

double dfs(int f) {
    if(dp[f]-EPS>0) {
        return dp[f];
    }
    for(int i=1;i<=n;++i) {
        if(f>c[i]) {
            dp[f]+=(int(TMP*c[i]*c[i]))*1.0/n;
        }
        else {
            dp[f]+=(1+dfs(f+c[i]))/n;;
        }
    }
    return dp[f];
}

int main() {
    while(scanf("%d%d",&n,&f)==2) {
        for(int i=1;i<=n;++i) {
            scanf("%lf",c+i);
        }
        memset(dp,0,sizeof(dp));
        printf("%.3lf\n",dfs(f));
    }
    return 0;
}