POJ 2096 Collecting Bugs (概率dp)

Collecting Bugs

Time Limit: 10000MS		Memory Limit: 64000K
Total Submissions: 3240		Accepted: 1601
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

题目链接： http://poj.org/problem?id=2096

题目大意： 一个软件有 s 个子系统， n 种 bug。某人一天能找到一个 bug。求在这个软件中找齐 n 种 bug，并且每个子系统中至少包含一个 bug 的时间的 期望

题目分析：基础的概率dp,dp[i][j]表示发现i种bug，j个子系统已发现bug离目标还需要的期望次数，首先dp[n][s]必然为0，分四种情况讨论，都没找到，找到新的i旧的j，找到旧的i新的j，找到新的i新的j，概率公式很好计算，方程从后往前递推即可

#include <cstdio>
#include <cstring>
int const MAX = 1005;
double dp[MAX][MAX];

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)
                    continue;
                dp[i][j] = (1.0 + ((1.0 * (n - i) / n) * (1.0 * j / s)) * dp[i + 1][j] +
                            ((1.0 * (s - j) / s) * (1.0 * i / n)) * dp[i][j + 1] +
                            ((1.0 * (n - i) / n) * (1.0 * (s - j) / s)) * dp[i + 1][j + 1])
                            / (1.0 - ((1.0 * i / n) * (1.0 * j / s)));
            }
        }
        printf("%.4f\n", dp[0][0]);
    }
}