POJ-2096-Collecting Bugs(概率DP)

Posted 2022-09-22 ydddd

链接：

https://vjudge.net/problem/POJ-2096#author=0

题意：

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.
某个系统中有n个子系统和m个bug类型，该系统每天会出现一个bug （属于某个子系统和某个bug类型），bug的类型是等概率的，bug也是等概率地出现在每个子系统的。问所有子系统都出现bug且所有的bug类型都出现的期望天数。

思路：

DP[i][j] 表示有i种bug，j个系统上有bug的天数期望。
由DP[i][j]可推出DP[i][j], DP[i+1][j], DP[i][j+1], DP[i+1][j+1].
概率分别为(ij)/(ns), (n-i)j/ns, i(s-j)/ns, (n-i)(s-j)/ns.
然后化简一下就行。

代码：

#include <iostream>
#include <memory.h>
#include <string>
#include <istream>
#include <sstream>
#include <vector>
#include <stack>
#include <algorithm>
#include <map>
#include <queue>
#include <math.h>
#include <cstdio>
#include <set>
#include <iterator>
#include <cstring>
#include <assert.h>
using namespace std;

typedef long long LL;
double Dp[1010][1010];

int main()

    int n, s;
    while(~scanf("%d%d", &n, &s))
    
        memset(Dp, 0, sizeof(Dp));
        Dp[n][s] = 0.0;
        for (int i = n; i >= 0; i--)
        
            for (int j = s; j >= 0; j--)
            
                if (i == n && j == s)
                    continue;
                Dp[i][j] = (n * s + (n - i) * j * Dp[i + 1][j] + i * (s - j) * Dp[i][j + 1] +
                            (n - i) * (s - j) * Dp[i + 1][j + 1]) / (n * s - i * j);
            
        
        printf("%.4lf\n", Dp[0][0]);
    

    return 0;