Score : $600$ points
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 $E$.
Input is given from Standard Input in the following format:
$N$ $M$ $s_1$ $t_1$ $:$ $s_M$ $t_M$
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 $10^{-6}$.
4 6 1 4 2 3 1 3 1 2 3 4 2 4
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 $E$.
3 2 1 2 2 3
2.0000000000
Blocking any one passage makes Takahashi unable to reach Room $N$, so Aoki cannot block a passage.
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
3.0133333333