LOOPS

Time Limit: 15000/5000 MS (Java/Others)    Memory Limit: 125536/65536 K (Java/Others)
Total Submission(s): 1651    Accepted Submission(s): 653


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

/*題意：有一個迷宮r行m列,開始點在[1,1]如今要走到[r,c]
對於在點[x,y]能夠開啟一扇門走到[x+1,y]或者[x,y+1]
消耗2點魔力
問平均消耗多少魔力能走到[r,c]

分析：如果dp[i][j]表示在點[i,j]到達[r,c]所須要消耗的平均魔力(期望)
則從dp[i][j]能夠到達:
dp[i][j],dp[i+1,j],dp[i][j+1];
相應概率分別為：
p1,p2,p3
由E(aA+bB+cC...)=aEA+bEB+cEC+...//包括狀態A,B,C的期望能夠分解子期望求解 
得到dp[i][j]=p1*dp[i][j]+p2*dp[i+1][j]+p3*dp[i][j+1]+2;
*/
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <string>
#include <queue>
#include <algorithm>
#include <map>
#include <cmath>
#include <iomanip>
#define INF 99999999
typedef long long LL;
using namespace std;

const int MAX=1000+10;
int n,m;
double dp[MAX][MAX],p[MAX][MAX][3];

int main(){
    while(~scanf("%d%d",&n,&m)){
        for(int i=1;i<=n;++i){
            for(int j=1;j<=m;++j)scanf("%lf%lf%lf",&p[i][j][0],&p[i][j][1],&p[i][j][2]);
        }
        memset(dp,0,sizeof dp);
        for(int i=n;i>=1;--i){
            for(int j=m;j>=1;--j){
                if(i == n && j == m)continue;
                if(p[i][j][0] == 1.00)continue;//該點無路可走,期望值肯定為0(dp[i][j]=0)
                dp[i][j]=(p[i][j][1]*(dp[i][j+1])+p[i][j][2]*(dp[i+1][j])+2)/(1-p[i][j][0]);
            }
        }
        printf("%.3lf\n",dp[1][1]);
    }
    return 0;
}