概率DP POJ 2096 Collecting Bugs

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本文探讨了一个程序中的Bug收集问题，通过数学期望的方法解决了一个复杂问题。问题要求在程序的多个子系统中收集所有类型的Bug，并计算完成这一任务所需的平均天数。

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题目链接：POJ 2096

Collecting Bugs

Time Limit: 10000MS		Memory Limit: 64000K
Total Submissions: 2202		Accepted: 1061
Case Time Limit: 2000MS		Special Judge

Description

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

Source

Northeastern Europe 2004, Northern Subregion

题意：

现在有一个程序，程序中有n种bug，每个bug可能在s个子程序中被发现，Ivan要每种bug至少找出一个，在每个子程序中至少找到一个。

问完成要求的时间期望（每天只找出一个bug）

思路：

求期望的问题，要转换思路倒推（可以先思考一下：一个人射箭，射中靶心的概率是0.6，问他第一次射中靶心时，射了多少次的期望？和如果他射了10次，他能射中靶心的期望？两个问题的不同）

参考kuangbin神的思路：

dp[i][j]表示已经找到i种bug,j个系统的bug，达到目标状态的天数的期望
         dp[n][s]=0;要求的答案是dp[0][0];
         dp[i][j]可以转化成以下四种状态:
              dp[i][j],发现一个bug属于已经有的i个分类和j个系统。概率为(i/n)*(j/s);
              dp[i][j+1],发现一个bug属于已有的分类，不属于已有的系统.概率为 (i/n)*(1-j/s);
              dp[i+1][j],发现一个bug属于已有的系统，不属于已有的分类,概率为 (1-i/n)*(j/s);
              dp[i+1][j+1],发现一个bug不属于已有的系统，不属于已有的分类,概率为 (1-i/n)*(1-j/s);

注意要整理之后，把dp[i][j]整理到等式一侧，得到方程才能求解。

代码：

#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<iostream>
using namespace std;
//dp[i][j]表示找到i种bug，在j个子程序中的期望天数,倒推得结果。
double dp[1005][1005]; 
int main(){
	int n,s;
	while(scanf("%d %d",&n,&s)!=EOF){
		memset(dp,0,sizeof(dp));
		for (int i = n; i >= 0; i--)
		{
			for (int j = s; j >= 0; j--)
			{
				if(i == n && j == s)dp[i][j] = 0.0;
				else{
					dp[i][j] = (dp[i+1][j+1]*(n-i)*(s-j)+dp[i][j+1]*(i)*(s-j)+dp[i+1][j]*(n-i)*(j)+n*s)/(n*s-i*j);
				}
			}
		}
		printf("%.4lf\n",dp[0][0]);
	}
	return 0;
}