hdu 3853 LOOPS (概率dp 逆推求期望)

题目链接

LOOPS

Time Limit: 15000/5000 MS (Java/Others)    Memory Limit: 125536/65536 K (Java/Others)
Total Submission(s): 2630    Accepted Submission(s): 1081

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：

题意：有一个迷宫r行m列,开始点在[1,1]现在要走到[r,c]
对于在点[x,y]可以打开一扇门走到[x+1,y]或者[x,y+1]
消耗2点魔力
问平均消耗多少魔力能走到[r,c]

分析：

和之前的题类似，逆推

d[i][j]表示从i行j列到r行c列的期望，d[i][j] = 1+d[i][j]*p1 + d[i][j+1]*p2 + d[i+1][j]*p3.

移项就是

 d[i][j] = (1.0 + d[i][j+1]*p[i][j].b + d[i+1][j]*p[i][j].c)/(1.0-p[i][j].a);

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <queue>
#include <cmath>
#include <algorithm>
#define LL __int64
const int maxn = 1e3 + 10;
using namespace std;
double d[maxn][maxn];
struct node
{
    double a, b, c;
}p[maxn][maxn];

int main()
{
    int r, c, i, j;
    while(~scanf("%d%d", &r, &c))
    {
        for(i = 1; i <= r; i++)
        for(j = 1; j <= c; j++)
        scanf("%lf%lf%lf", &p[i][j].a, &p[i][j].b, &p[i][j].c);

        memset(d, 0, sizeof(d));
        d[r][c] = 0;
        for(i = r; i >= 1; i--)
        for(j = c; j >= 1; j--)
        if(i==r && j==c)
        continue;
        else if(p[i][j].a - 1.0==0)  //因为这错了两次，如果这个点一只是在原地的话，逆推是不可达的点，也就是d[i][j]=0.
        continue;
        else
        d[i][j] = (1.0 + d[i][j+1]*p[i][j].b + d[i+1][j]*p[i][j].c)/(1.0-p[i][j].a);
        printf("%.3lf\n", 2.0*d[1][1]);
    }
    return 0;
}