Robberies小偷（01背包变形）

题目描述

The aspiring Roy the Robber has seen a lot of American movies, and knows that the bad guys usually gets caught in the end, often because they become too greedy. He has decided to work in the lucrative business of bank robbery only for a short while, before retiring to a comfortable job at a university.

For a few months now, Roy has been assessing the security of various banks and the amount of cash they hold. He wants to make a calculated risk, and grab as much money as possible.


His mother, Ola, has decided upon a tolerable probability of getting caught. She feels that he is safe enough if the banks he robs together give a probability less than this.

输入

The first line of input gives T, the number of cases. For each scenario, the first line of input gives a floating point number P, the probability Roy needs to be below, and an integer N, the number of banks he has plans for. Then follow N lines, where line j gives an integer Mj and a floating point number Pj . 
Bank j contains Mj millions, and the probability of getting caught from robbing it is Pj .

输出

For each test case, output a line with the maximum number of millions he can expect to get while the probability of getting caught is less than the limit set.

Notes and Constraints
0 < T <= 100
0.0 <= P <= 1.0
0 < N <= 100
0 < Mj <= 100
0.0 <= Pj <= 1.0
A bank goes bankrupt if it is robbed, and you may assume that all probabilities are independent as the police have very low funds.

样例输入

30.04 31 0.022 0.033 0.050.06 32 0.032 0.033 0.050.10 31 0.032 0.023 0.05

样例输出

246

分析：

01背包的概率问题

当前的概率基于前一种状态的概率，即偷n家银行而不被抓的概率等于偷n-1家银行不被转的概率乘以偷第n家银行不被抓的概率。

  用dp[i]表示偷价值为 i 时不被抓的概率，则状态转移方程为：

  dp[j] = max(dp[j] , dp[j-m[i]] * (1-p[i]));

  自己写关键在01背包的转换，原意是提供银行个数和期望被捕概率，然后将每个银行的钱数和逃脱概率给出，

  通过将总数当作背包大小，通过求最大逃脱概率当作最大价值（但是并不是求这个），最终通过从总钱数递减找到低于期望被捕概率第一项背包，

  即为不被逮捕的所能强盗的最大钱数。

#include <iostream>#include <stdio.h>#include <cstring>#include <algorithm>using namespace std;const int maxn=101;const int maxv=10005;float dp[maxv];  //求最大的不被抓概率float p; //被抓的概率要<=pint n;int sum;  //银行的总钱数int money[maxn];float prob[maxn];int main(){    int t,val;    float pj;    scanf("%d",&t);    while(t--)    {        scanf("%f%d",&p,&n);        sum=0;        for(int i=1; i<=n; i++)        {            scanf("%d%f",&val,&pj);            sum+=val;//算出总的钱数            money[i]=val;  //将银行里的钱当成容量            prob[i]=1-pj;   //将不被抓的可能性当成价值        }        memset(dp,0,sizeof(dp));        dp[0]=1;//偷0元不被抓的概率为1        for(int i=1; i<=n; i++)            for(int j=sum; j>=money[i]; j--)//选择偷与不偷                dp[j]=max(dp[j],dp[j-money[i]]*prob[i]);        int ans=0; //别忘了初始为0        for(int i=sum; i>=1; i--)//找出不被抓的最大概率            if(dp[i]>=1-p)            {                ans=i;                break;            }        printf("%d\n",ans);    }    return 0;}