An undirected connected graph with vertices numbered from 1 to N and edges numbered from 1 to M.  
Small Z walks randomly on the graph. At the beginning, small Z is at vertex 1. In each step, small Z randomly selects an edge of the current vertex with equal probability, walks along this edge to the next vertex, and obtains a fraction equal to the number of this edge. When small Z reaches the N vertex, the walk ends, and the total score is the sum of all the scores obtained.  
Now, please number the M-edges to minimize the expected total score of small Z.

The first row is a positive integer N and M, respectively, representing the number of vertices and edges of the graph. Next, each row of row M is an integer u,v(1 ≤ u,v ≤ N), indicating that there is an edge between vertex u and vertex v. The input ensures that 30% of the data satisfies N ≤ 10100% of the data satisfies 2 ≤ N ≤ 500 and is a simple undirected connected graph.

Contains only one real number, representing the minimum expected value, with 3 decimal places reserved.

Unofficial data

#include<cstdio>
#include<cstring>
#include<algorithm>
#define getchar() (p1==p2&&(p2=(p1=buf)+fread(buf,1,1<<23,stdin),p1==p2)?EOF:*p1++)
using namespace std;
const int MAXN=1e6+10;
const double eps=1e-7;
char buf[1<<23],*p1=buf,*p2=buf;
inline int read()
{
    char c=getchar();int x=0,f=1;
    while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){x=x*10+c-'0';c=getchar();}
    return x*f;
}
int N,M;
struct node
{
    int u,v,nxt;
}edge[MAXN];
int head[MAXN],num=1;
inline void AddEdge(int x,int y)
{
    edge[num].u=x;
    edge[num].v=y;
    edge[num].nxt=head[x];
    head[x]=num++;
}
double f[1001][1001],ans[MAXN],E[MAXN],inder[MAXN];
int S[MAXN],T[MAXN];
int dcmp(double x)
{
    if(x<eps&&x>-eps) return 0;
    else return x<0?-1:1;
}
void Gauss()
{

    for(int i=1;i<N;i++)
    {
        int mx=i;
        for(int j=i+1;j<N;j++)
            if( dcmp(f[j][i]-f[mx][i])>0 ) mx=j;
        if(mx!=i) swap(f[i],f[mx]);
        for(int j=i+1;j<N;j++)
        {
            double tmp=f[j][i]/f[i][i];
            for(int k=i;k<=N;k++)
                f[j][k]-=(double)tmp*f[i][k];
        }
    }
    for(int i=N-1;i>=1;i--)
    {
        for(int j=i+1;j<N;j++)
            f[i][N]-=ans[j]*f[i][j];
        ans[i]=f[i][N]/f[i][i];
    }
}
int main()
{
    #ifdef WIN32
    freopen("a.in","r",stdin);
    #else
    #endif
    memset(head,-1,sizeof(head));
    N=read(),M=read();
    for(int i=1;i<=M;i++)
    {
        int x=read(),y=read();
        AddEdge(x,y);AddEdge(y,x);
        inder[x]++;inder[y]++;
        S[i]=x;T[i]=y;
    }
    f[1][N]=1;
    for(int i=1;i<N;i++) f[i][i]=1;
    for(int i=1;i<N;i++)
        for(int j=head[i];j!=-1;j=edge[j].nxt)
            if(edge[j].v!=N)
                f[i][edge[j].v]=(double)-1.00/inder[edge[j].v];
    Gauss();
    for(int i=1;i<=M;i++) 
        E[i]=ans[S[i]]/inder[S[i]]+ans[T[i]]/inder[T[i]];
    sort(E+1,E+M+1);
    double out=0;
    for(int i=1;i<=M;i++)
        out+=E[i]*(M-i+1);
    printf("%.3lf",out);
    return 0;
}