Problem:
Score : 600 points

Problem Statement

There is a cave consisting of N rooms and M one-directional passages. The rooms are numbered 1 through N.

Takahashi is now in Room 1, and Room N has the exit. The i-th passage connects Room s_i and Room t_i (s_i < t_i) and can only be traversed in the direction from Room s_i to Room t_i. It is known that, for each room except Room N, there is at least one passage going from that room.

Takahashi will escape from the cave. Each time he reaches a room (assume that he has reached Room 1 at the beginning), he will choose a passage uniformly at random from the ones going from that room and take that passage.

Aoki, a friend of Takahashi's, can block one of the passages (or do nothing) before Takahashi leaves Room 1. However, it is not allowed to block a passage so that Takahashi is potentially unable to reach Room N.

Let E be the expected number of passages Takahashi takes before he reaches Room N. Find the value of E when Aoki makes a choice that minimizes E.

Constraints

2 \leq N \leq 600

N-1 \leq M \leq \frac{N(N-1)}{2}

s_i < t_i

If i != j, (s_i, t_i) \neq (s_j, t_j). (Added 21:23 JST)

For every v = 1, 2, ..., N-1, there exists i such that v = s_i.

Input

Input is given from Standard Input in the following format:

N M
s_1 t_1
:
s_M t_M

Output

Print the value of E when Aoki makes a choice that minimizes E.
Your output will be judged as correct when the absolute or relative error from the judge's output is at most 10^{-6}.

Sample Input 1

4 6
1 4
2 3
1 3
1 2
3 4
2 4

Sample Output 1

1.5000000000

If Aoki blocks the passage from Room 1 to Room 2, Takahashi will go along the path 1 → 3 → 4 with probability \frac{1}{2} and 1 → 4 with probability \frac{1}{2}. E = 1.5 here, and this is the minimum possible value of E.

Sample Input 2

3 2
1 2
2 3

Sample Output 2

2.0000000000

Blocking any one passage makes Takahashi unable to reach Room N, so Aoki cannot block a passage.

Sample Input 3

10 33
3 7
5 10
8 9
1 10
4 6
2 5
1 7
6 10
1 4
1 3
8 10
1 5
2 6
6 9
5 6
5 8
3 6
4 8
2 7
2 9
6 7
1 2
5 9
6 8
9 10
3 9
7 8
4 5
2 10
5 7
3 5
4 7
4 9

Sample Output 3

3.0133333333

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;
public static class P {
  public static void Main() {
    var nm = Console.ReadLine().Split().Select(int.Parse).ToArray();
    int n = nm[0];
    int m = nm[1];
    var graph = new List< List < int > >();
    foreach (int _ in Enumerable.Range(0, n)) {
      graph.Add(new List< int >());
    }
    foreach (int _ in Enumerable.Range(0, m)) {
      var st = Console.ReadLine().Split().Select(int.Parse).ToArray();
      int s = st[0] - 1;
      int t = st[1] - 1;
      graph[s].Add(t);
    }

    var dp = new List < double >();
    dp.AddRange(Enumerable.Repeat(0.0, n));
    foreach (int i in Enumerable.Range(0, n - 1).Reverse()) {
      foreach (int j in graph[i]) {
        dp[i] += dp[j];
      }
      dp[i] /= graph[i].Count();
      dp[i] += 1;
    }

    var ep = new List < double >();
    ep.AddRange(Enumerable.Repeat(0.0, n));
    ep[0] = 1;
    foreach (int i in Enumerable.Range(0, n)) {
      foreach (int j in graph[i]) {
        ep[j] += ep[i] / graph[i].Count();
      }
    }

    double full = dp[0];
    double ans = full;
    foreach (int i in Enumerable.Range(0, n)) {
      int sz = graph[i].Count();
      if (sz < 2) continue;
      double sum = 0;
      double max = 0;
      foreach (int j in graph[i]) {
        sum += dp[j];
        max = Max(max, dp[j]);
      }
      double x = sum / sz;
      double y = (sum - max) / (sz - 1);
      ans = Min(ans, full - ep[i] * (x - y));
    }
    Console.WriteLine(ans);
  }
}