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
From the perspective of expected DP, contrary to probability DP, it is generally recursive from back to front, so for each throw of the sieve, its expectation is the average value of all expectations from +1 to +6+1 (including itself), But this question has a flight route, which can jump directly from one point to another. Then we can store these routes in a tree. For each point, if there is a corresponding node, then this point Will change the expected value DP of the corresponding point on the same route to the same value.
Then mark the next node of the chain, indicating that this point has been passed and does not need to be calculated again
#include <stdio.h>
#include <string.h>
#include <math.h>
#include <map>
#include <stack>
#include <set>
#include <queue>
#include <vector>
#include <iostream>
#include <algorithm>
#include <cstring>
#include <string>
#include <cstdio>
#define ls (p<<1)
#define rs (p<<1|1)
//#include<bits/stdc++.h>
using namespace std;
const int maxn=1e6;
bool father[maxn];
double dp[maxn];
int a[maxn],vis[maxn];
vector<int>edge[maxn];
void solve(){
int n,m;
while(scanf("%d%d",&n,&m)!=EOF){
if(n==0&&m==0) break;
for(int i=0;i<=n;i++)
edge[i].clear();
memset(vis,0,sizeof(vis));
for(int i=1;i<=m;i++){
int a,b;
cin>>a>>b;
edge[b].push_back(a);
edge[a].push_back(b);
}
for(int i=0;i<edge[n].size();i++){
vis[edge[n][i]]=1;
}
memset(dp,0,sizeof(dp));
for(int i=n-1;i>=0;i--){
if(vis[i]==0){
double ans=0;
for(int j=1;j<=6;j++)
ans+=dp[i+j]/6;
dp[i]=ans+1;
vis[i]=1;
}
for(int j=0;j<edge[i].size();j++){
dp[edge[i][j]]=dp[i];
vis[edge[i][j]]=1;
}
}
printf("%.4lf\n",dp[0]);
}
}
int main()
{
ios::sync_with_stdio(false),cin.tie(0),cout.tie(0);
solve();
return 0;
}