Collecting Bugs(POJ 2096)

本文探讨了在一个拥有多个子系统和多种分类的软件中,收集到所有子系统和分类的bug所需平均天数的算法实现。通过动态规划方法,详细解析了不同状态转移的概率及其对期望天数的影响。

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一个软件有s个子系统,会产生n种bug。某人一天发现一个bug,这个bug属于一个子系统,属于一个分类。每个bug属于某个子系统的概率是1/s,属于某种分类的概率是1/n 。则每个子系统中找到至少一个bug,并且每个类别至少有一个bug所需天数的期望?

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.

反解dp[i][j]可得

 

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)
#include <iostream>
#include <stdio.h>

using namespace std;

#define MAXN 1002
int N;//类别数

int S;//系统数

double dp[MAXN][MAXN]; //1001+1

double solve() {
    for (int i = N; i >= 0; i--) {
        for (int j = S; j >= 0; j--) {
            if (i == N and j == S) {
                continue;
            }
            dp[i][j] = ((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) / (N * S - i * j);
        }
    }
    return dp[0][0];
}

int main() {
    cin >> N >> S;
    printf("%.4lf\n", solve());
}

 

s is a fascinating hobby for many people, particularly those interested in studying insects and their behavior. However, it's crucial to follow ethical and legal guidelines, and to ensure that bugs are collected and handled humanely. Here are some tips for collecting bugs: 1. Check local regulations: Before you start collecting bugs, check your local regulations to make sure it is legal to do so. Some areas may have restrictions on collecting certain species or collecting bugs in protected areas. 2. Use ethical and humane methods: It's important to use ethical and humane methods when collecting bugs. Avoid using harmful chemicals, and instead use gentle methods such as nets or traps. Handle the bugs carefully and avoid injuring them. 3. Respect the environment: When collecting bugs, make sure to respect the environment and the other creatures that live there. Avoid damaging plants or other habitats, and leave the area as you found it. 4. Keep accurate records: Keep accurate records of the species you collect, when and where you collected them, and any other relevant information. This can be helpful for scientific research and for tracking changes in insect populations over time. 5. Release the bugs: After you have collected and observed the bugs, release them back into their natural habitat. Make sure to do so in a safe and appropriate location, and handle them carefully to avoid injury. Remember that collecting bugs can be a fun and educational hobby, but it's important to do so responsibly and ethically. By following these guidelines, you can enjoy the hobby while also helping to protect these fascinating creatures and their habitats.
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