Aeroplane chess

Time Limit: 1 Sec Memory Limit: 32 MB

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.
　　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

HINT

1≤N≤100000, 0≤M≤1000

Main idea

从0走到n-1，每次以均等概率走1~6步，某些点可以直接跨越到指定点，求走出n所需走的次数的期望。

Solution

我们直接使用期望DP求解。令 f[i] 表示到位置 i 的期望，然后直接从后往前递推即可。

Code

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#include<bits/stdc++.h>
using namespace std;
const int ONE = 100001;

int n,m;
int x,y,To[ONE];
double f[ONE];

int get()
{
    int res=1,Q=1;  char c;
    while( (c=getchar())<48 || c>57)
        if(c=='-')Q=-1;
    if(Q) res=c-48;
    while((c=getchar())>=48 && c<=57)
        res=res*10+c-48;
    return res*Q;
}

void Solve()
{
    n=get();    m=get();
    if(!n && !m) exit(0);
    memset(To,0,sizeof(To)); memset(f,0,sizeof(f));
    for(int i=1;i<=m;i++)
    {
        x=get();    y=get();
        To[x] = y;
    }

    for(int i=n-1;i>=0;i--)
    {
        if(To[i]) {f[i] = f[To[i]]; continue;}
        for(int j=1;j<=6;j++)
            f[i] += (double)1/6 * f[min(i+j,n)];
        f[i]++;
    }

    printf("%.4lf\n",f[0]);
}

int main()
{
    for(;;)
        Solve();
}