文章标题

Collecting Bugs

Time Limit: 10000MS Memory Limit: 64000K

Total Submissions: 1245 Accepted: 524

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

简单期望dp的一类，题目大意是：有n个软件，和s种bug，这个人每天可以找一个bug，这个bug属于这n个软件之一。问，找完所有n个机器，s种bug的期望

最开始做这道题的时候，网上搜了好多傻逼博客，上面只是给出了只找一天的期望，然后代码里面确是计算了找一天两天，无数天的期望，太蠢，不忍直视，于是写下这篇博客。

设dp[i][j]表示 已经找到了j种bug，且这j种bug属于i个软件的需要的天数，那么有以下四种转移

dp[i][j] = dp[i+1][j+1] * (n-i) * (s-j) / n / s;

dp[i][j] = dp[i][j+1] * i * (s-j) / n / s;

dp[i][j] = dp[i+1][j] * (n-i) * j / n / s;

接下来是重点

dp[i][j] = dp[i][j] * i * j / n / s;

这就是找一天能转移到达的状态。

之前三个好理解，

我们可以由第四个方程，利用等比数列极限求和得到转移无数天之后的答案就是:dp[i][j] = dp[i][j] / (1 - i*j/n/s);

（dp[i][j]根据前三个公式算出来一天的情况，再用等比数列求和计算无限天的情况）

代码自己写。