Aeroplane chess

Problem 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

概率DP主要用于求解期望、概率等题目。

一般求概率是正推，求期望是逆推。

kuangbin的概率DP学习网址：http://www.cnblogs.com/kuangbin/archive/2012/10/02/2710606.html

 1 #include <cstdio>
 2 #include <cstring>
 3 #include <algorithm>
 4 #include <vector>
 5 using namespace std;
 6 #define N 100010
 7 
 8 double dp[N];
 9 int nxt[N];
10 
11 int main()
12 {
13     int n, m;
14     while(~scanf("%d%d", &n, &m), n+m) {
15         memset(nxt, -1, sizeof(nxt));
16         for(int i = 0; i < m; i++) {
17             int u, v;
18             scanf("%d%d", &u, &v);
19             nxt[u] = v;
20         }
21         memset(dp, 0, sizeof(dp));
22         double dec = (double)1 / 6;
23         for(int i = n - 1; i >= 0; i--) {
24             if(nxt[i] != -1) {
25                 dp[i] = dp[nxt[i]]; //如果可以飞，就直接把上一步的值赋给它
26                 continue;
27             }
28             for(int j = 1; j <= 6; j++) {
29                 if(i + j <= n) {
30                     dp[i] += dp[i + j] * dec; //不能飞的话，就掷骰子为1-6的概率都为1/6，递推
31                 }
32             }
33             dp[i]++; //走到下一步要+1
34         }
35         printf("%.4f
", dp[0]);
36     }
37     return 0;
38 }