hdu 4405 Aeroplane chess 概率dp入门题

Description

Hzz loves aeroplane chess very much. The chess map contains N+1 grids labeled from 0 to N. Hzz starts at grid 0. For each step he throws a dice(a dice have six faces with equal probability to face up and the numbers on the faces are 1,2,3,4,5,6). When Hzz is at grid i and the dice number is x, he will moves to grid i+x. Hzz finishes the game when i+x is equal to or greater than N.
There are also M flight lines on the chess map. The i-th flight line can help Hzz fly from grid Xi to Yi (0<Xi<Yi<=N) without throwing the dice. If there is another flight line from Yi, Hzz can take the flight line continuously. It is granted that there is no two or more flight lines start from the same grid.
Please help Hzz calculate the expected dice throwing times to finish the game.

Input

There are multiple test cases.
Each test case contains several lines.
The first line contains two integers N(1≤N≤100000) and M(0≤M≤1000).
Then M lines follow, each line contains two integers Xi,Yi(1≤Xi<Yi≤N).  
The input end with N=0, M=0.

Output

For each test case in the input, you should output a line indicating the expected dice throwing times. Output should be rounded to 4 digits after decimal point.

Sample Input

2 0
8 3
2 4
4 5
7 8
0 0 

Sample Output

1.1667
2.3441 

 

题目大意：有n+1个格子排成一排，编号从0到n，并且在某些格子能直接跳到某个格子。

现在掷骰子，掷到多少就走几步，求掷骰子次数的期望。

 

期望：$E(x) = \sum  {p_i*x_i}$

 

定义dp[i]是从i到达终点的掷骰子期望次数，那么$$dp[i] = \sum_{j=1}^6 {(dp[i+j]+1)/6}$$。

 

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
using namespace std;
const int maxn = 100010;
int jump[maxn];
double dp[maxn];
int main()
{
    int n,m,a,b;
    while(scanf("%d%d",&n,&m)!=EOF&&n) {
        memset(jump,0,sizeof(jump));
        memset(dp,0,sizeof(dp));
        while(m--) {
            scanf("%d%d",&a,&b);
            jump[a] = b;
        }
        dp[n] = 0;
        for(int i=n-1;i>=0;i--) {
            if(jump[i]) dp[i] = dp[jump[i]];
            else
            for(int j=1;j<=6;j++) {
                dp[i] += (dp[i+j]+1)/6;
            }
        }
        printf("%.4f\n",dp[0]);
    }
}