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

 

題目大意：一個軟件有 s 個子系統， n 種 bug。某人一天能找到一個 bug。求在這個軟件中找齊 n 種 bug，並且每個子系統中至少包含一個 bug 的時間的期望

題目分析：基礎的概率dp,dp[i][j]表示發現i種bug，j個子系統已發現bug離目標還需要的期望次數，首先dp[n][s]必然為0，分四種情況討論，都沒找到，找到新的i舊的j，找到舊的i新的j，找到新的i新的j，概率公式很好計算，方程從後往前遞推即可

#include 
#include 
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
, dp[0][0]);
    }
}