HDU3853：LOOPS（概率dp & 期望）

LOOPS

Time Limit: 15000/5000 MS (Java/Others)    Memory Limit: 125536/65536 K (Java/Others)
Total Submission(s): 6443    Accepted Submission(s): 2583

Problem Description

Akemi Homura is a Mahou Shoujo (Puella Magi/Magical Girl).

Homura wants to help her friend Madoka save the world. But because of the plot of the Boss Incubator, she is trapped in a labyrinth called LOOPS.

The planform of the LOOPS is a rectangle of R*C grids. There is a portal in each grid except the exit grid. It costs Homura 2 magic power to use a portal once. The portal in a grid G(r, c) will send Homura to the grid below G (grid(r+1, c)), the grid on the right of G (grid(r, c+1)), or even G itself at respective probability (How evil the Boss Incubator is)!
At the beginning Homura is in the top left corner of the LOOPS ((1, 1)), and the exit of the labyrinth is in the bottom right corner ((R, C)). Given the probability of transmissions of each portal, your task is help poor Homura calculate the EXPECT magic power she need to escape from the LOOPS.

 

Input

The first line contains two integers R and C (2 <= R, C <= 1000).

The following R lines, each contains C*3 real numbers, at 2 decimal places. Every three numbers make a group. The first, second and third number of the cth group of line r represent the probability of transportation to grid (r, c), grid (r, c+1), grid (r+1, c) of the portal in grid (r, c) respectively. Two groups of numbers are separated by 4 spaces.

It is ensured that the sum of three numbers in each group is 1, and the second numbers of the rightmost groups are 0 (as there are no grids on the right of them) while the third numbers of the downmost groups are 0 (as there are no grids below them).

You may ignore the last three numbers of the input data. They are printed just for looking neat.

The answer is ensured no greater than 1000000.

Terminal at EOF

 

Output

A real number at 3 decimal places (round to), representing the expect magic power Homura need to escape from the LOOPS.

 

Sample Input

2 20.00 0.50 0.50    0.50 0.00 0.500.50 0.50 0.00    1.00 0.00 0.00

 

Sample Output

6.000

 

Source

2011 Invitational Contest Host by BUPT

 

题意：一个N*M的房间，主角在1，1点，每次使用2个魔法可以触发当前房间的传送门，每个房间有概率p1 p2  p3分别是停留在原地，进入右边的房间，进入下边的房间，问到达N M点的期望魔法数。

思路：易推得dp(x,y) = p1dp(x,y)+p2dp(x+1,y)+p3dp(x,y+1)+2;
dp(x,y)= (p2dp(x+1,y)+p3dp(x,y+1)+2)/(1-p1); 还有一种情况，停在原地的概率为1的话，这个房间跳过即可。

# include <iostream># include <cstdio># include <cmath># include <cstring>using namespace std;const int maxn = 1e3+8;double a[maxn][maxn], b[maxn][maxn], c[maxn][maxn], dp[maxn][maxn];int main(){    int n, m;    while(~scanf("%d%d",&n,&m))    {        memset(dp, 0, sizeof(dp));        for(int i=1; i<=n; ++i)            for(int j=1; j<=m; ++j)                scanf("%lf%lf%lf",&a[i][j],&b[i][j],&c[i][j]);        dp[n][m] = 0;        for(int i=n; i>0; --i)        {            for(int j=m; j>0; --j)            {                if(i==n&&j==m) continue;                if(fabs(1-a[i][j])<1e-7) continue;//进入了这点就走不出去。                dp[i][j] = (b[i][j]*dp[i][j+1]+c[i][j]*dp[i+1][j]+2)/(1-a[i][j]);            }        }        printf("%.3f\n",dp[1][1]);    }    return 0;}