Fork in the Road

AtCoder

IDabc144_f

Time2000ms

Memory256MB

Difficulty—

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: $N$ $M$ $s_1$ $t_1$ $:$ $s_M$ $t_M$ [samples]

Samples

Input #1

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 $E$.

Input #2

3 2
1 2
2 3

Output #2

2.0000000000

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

Input #3

Output #3

3.0133333333

API Response (JSON)

{
  "problem": {
    "name": "Fork in the Road",
    "description": {
      "content": "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 ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc144_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a cave consisting of $N$ rooms and $M$ one-directional passages. The rooms are numbered $1$ through $N$.\nTakahashi is now in Room $1$, and Room $N$ has the exit. The $i$\\-th passage connects ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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