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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;

public class Main {
	public static void main(String[] args) throws Exception {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
		String[] sa = br.readLine().split(" ");
		int n = Integer.parseInt(sa[0]);
		int m = Integer.parseInt(sa[1]);
		List<Node> list = new ArrayList<>(n);
		for (int i = 0; i < n; i++) {
			list.add(new Node(i));
		}
		for (int i = 0; i < m; i++) {
			sa = br.readLine().split(" ");
			int s = Integer.parseInt(sa[0]) - 1;
			int t = Integer.parseInt(sa[1]) - 1;
			list.get(s).list.add(new Hen(t));
		}
		br.close();

		for (int i = n - 2; i >= 0; i--) {
			Node node = list.get(i);
			double sum = 0;
			for (Hen h : node.list) {
				Node next = list.get(h.v);
				h.c = next.exp + 1;
				sum += h.c;
			}
			node.exp = sum / node.list.size();
		}
		double ans = list.get(0).exp;

		for (int i = 0; i < n - 1; i++) {
			Collections.sort(list.get(i).list);
		}

		for (int i = 0; i < n - 1; i++) {
			for (int j = n - 2; j >= 0; j--) {
				Node node = list.get(j);
				int start = 0;
				if (node.v == i) {
					if (node.list.size() <= 1) {
						continue;
					}
					start++;
				}
				double sum = 0;
				for (int k = start; k < node.list.size(); k++) {
					Hen h = node.list.get(k);
					Node next = list.get(h.v);
					h.c = next.exp + 1;
					sum += h.c;
				}
				node.exp = sum / (node.list.size() - start);
			}
			ans = Math.min(ans, list.get(0).exp);
		}
		System.out.println(ans);
	}

	static class Hen implements Comparable<Hen> {
		int v;
		double c;

		public Hen(int v) {
			this.v = v;
		}
		public int compareTo(Hen o) {
			return Double.compare(o.c, c);
		}
	}

	static class Node {
		int v;
		double exp;
		List<Hen> list = new ArrayList<>();

		public Node(int v) {
			this.v = v;
		}
	}
}