HDU3853:LOOPS(概率DP)

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 2
0.00 0.50 0.50    0.50 0.00 0.50
0.50 0.50 0.00    1.00 0.00 0.00

Sample Output

6.000

题意：迷宫是一个R*C的布局，每个格子中给出停留在原地，往右走一个，往下走一格的概率，起点在(1,1)，终点在(R,C)，每走一格消耗两点能量，求出最后所需要的能量期望

思路：期望DP的入门，dp[i][j]记录从i,j出发到终点所需期望，map[i][j][k]，记录每格三种状况的概率，按照数学期望公式去计算即可

#include <stdio.h>
#include <string.h>
#include <math.h>
#include <algorithm>
using namespace std;

double dp[1005][1005],map[1005][1005][3];

int main()
{
    int r,c,i,j,k;
    while(~scanf("%d%d",&r,&c))
    {
        memset(dp,0,sizeof(dp));
        for(i = 1; i<=r; i++)
            for(j = 1; j<=c; j++)
                for(k = 0; k<3; k++)
                    scanf("%lf",&map[i][j][k]);
        for(i = r; i>0; i--)
        {
            for(j = c; j>0; j--)
            {
                if(i == r && j == c)//到达终点了
                    continue;
                if(fabs(1-map[i][j][0])<1e-7)//停留原地的概率为1
                    continue;
                dp[i][j] = (map[i][j][1]*dp[i][j+1]+map[i][j][2]*dp[i+1][j]+2)/(1-map[i][j][0]);//+2是到下一格消耗的能量
            }
        }
        printf("%.3lf\n",dp[1][1]);
    }

    return 0;
}