POJ2096Collecting Bugs（数学期望，概率DP）

Posted 2020-10-17 ---学习ing---

问题：

Ivan is fond of collecting. Unlike other people who collect post stamps, coins or other material stuff, he collects software bugs. When Ivan gets a new program, he classifies all possible bugs into n categories. Each day he discovers exactly one bug in the program and adds information about it and its category into a spreadsheet. When he finds bugs in all bug categories, he calls the program disgusting, publishes this spreadsheet on his home page, and forgets completely about the program. 
Two companies, Macrosoft and Microhard are in tight competition. Microhard wants to decrease sales of one Macrosoft program. They hire Ivan to prove that the program in question is disgusting. However, Ivan has a complicated problem. This new program has s subcomponents, and finding bugs of all types in each subcomponent would take too long before the target could be reached. So Ivan and Microhard agreed to use a simpler criteria --- Ivan should find at least one bug in each subsystem and at least one bug of each category. 
Macrosoft knows about these plans and it wants to estimate the time that is required for Ivan to call its program disgusting. It‘s important because the company releases a new version soon, so it can correct its plans and release it quicker. Nobody would be interested in Ivan‘s opinion about the reliability of the obsolete version. 
A bug found in the program can be of any category with equal probability. Similarly, the bug can be found in any given subsystem with equal probability. Any particular bug cannot belong to two different categories or happen simultaneously in two different subsystems. The number of bugs in the program is almost infinite, so the probability of finding a new bug of some category in some subsystem does not reduce after finding any number of bugs of that category in that subsystem. 
Find an average time (in days of Ivan‘s work) required to name the program disgusting.

Input

Input file contains two integer numbers, n and s (0 < n, s <= 1 000).

Output

Output the expectation of the Ivan‘s working days needed to call the program disgusting, accurate to 4 digits after the decimal point.

Sample Input

1 2

Sample Output

3.0000

 题意：

有s个系统,有n种bug,bug的数量不限,一位程序员每天可以发现一个bug 现在求发现n种bug存在s个系统中并且每个系统都要被发现bug的平均天数(期望) 

即，一个n*s的矩阵，每天任选一个格子上涂一笔（可以重复涂），求达到每一行每一列都至少被涂了一笔的天数期望。

假设一个矩阵每一行每一列至少一个点被涂，我们称这个矩阵被覆盖。

思路：

从后向前推。

dp[i][j]表示在以及覆盖了一个i*j的矩阵的情况下要达到覆盖n*m的矩阵的天数期望。

dp[i][j]=dp[i][j+1]*p1+dp[i+1][j]*p2+dp[i+1][j+1]*p3+dp[i][j]*p4+1;

概率dp，公式自己推。

疑问：

为什么输出时".lf"WA 了，然而".f"AC了，这个影响精度吗。

#include<cstdio>
#include<cstdlib>
#include<iostream>
using namespace std;
double dp[1010][1010];
int main()
{
    int n,s,i,j;
    while(~scanf("%d%d",&n,&s)){
        dp[n][s]=dp[n+1][s]=dp[n][s+1]=dp[n+1][s+1] =0;
        for(i=n;i>=0;i--)
         for(j=s;j>=0;j--){
             if(i==n&&j==s) continue;
             dp[i][j]=(i*(s-j)*dp[i][j+1]+(n-i)*j*dp[i+1][j]+(n-i)*(s-j)*dp[i+1][j+1]+n*s)/(n*s-i*j);
         }
        printf("%.4f\n",dp[0][0]);
    }
    return 0;
}