POJ 2096 Collecting Bugs (probabilistic 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 with all bugs categories, he calls the program disgusting, publishes this spreadsheet on his home page, an D forgets completely about the program.
Companies, Macrosoft and Microhard is in tight competition. Microhard wants to decrease sales of one Macrosoft program. They hire Ivan to prove, the program in question is disgusting. However, Ivan has a complicated problem. This new program have S subcomponents, and finding bugs of all types in each subcomponent would take too long before the TA Rget could be reached. So Ivan and Microhard agreed to use a simpler criteria---Ivan should find at least one bugs in each subsystem and at LEAs T one bug of each category.
Macrosoft knows about these plans and it wants to estimate the time that's required for Ivan to call their program Disgusti Ng. It's important because the company releases a new version of soon, so it can correct its plans and release it quicker. Nobody would is interested in Ivan's opinion about the reliability of the obsolete version.
A Bug found in the program can is of any category with equal probability. Similarly, the bug can be found in any given subsystem with equal probability. Any particular bugs cannot belong to the different categories or happen simultaneously in both different subsystems. The number of bugs in the program was almost infinite, so the probability of finding a new bug of some category in some sub System does not reduce after finding any number of bugs of the that category in the that subsystem.
Find an average time (in days of Ivan's work) required to name the program disgusting.

Input

Input file contains the 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-4 digits after the de Cimal Point.

Sample Input

1 2

Sample Output

3.0000

Source

Northeastern Europe 2004, Northern subregion

Title Link: http://poj.org/problem?id=2096

Main topic:A software has s subsystem, n kinds of bugs. Someone can find a bug in one day. Gathered up n bugs in this software, and the time at least one bug in each subsystemexpected

Topic Analysis: The basis of probability dp,dp[i][j] indicates that I BUG,J subsystem has found that the bug from the target also need to expect the number of times, first Dp[n][s] must be 0, divided into four kinds of situation discussion, have not found, find new I old J, find the old i new J, find new I new J, The probability formula is very good calculation, the equation from the backward forward recursion can

#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)                    Continu e;                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]);}    }

Copyright NOTICE: This article for Bo Master original article, without Bo Master permission not reproduced.

POJ 2096 Collecting Bugs (probabilistic DP)