Rush Hour 2

AtCoder

IDabc204_e

Time2000ms

Memory256MB

Difficulty—

The Republic of AtCoder has $N$ cities and $M$ roads. The cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ connects City $A_i$ and City $B_i$ bidirectionally. There is a rush hour in the country that peaks at time $0$. If you start going through Road $i$ at time $t$, it will take $C_i+ \left\lfloor \frac{D_i}{t+1} \right\rfloor$ time units to reach the other end. ($\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) Takahashi is planning to depart City $1$ at time $0$ or some **integer time** later and head to City $N$. Print the earliest time when Takahashi can reach City $N$ if he can stay in each city for an **integer number of time units**. It can be proved that the answer is an integer under the Constraints of this problem. If City $N$ is unreachable, print `-1` instead. ## Constraints * $2 \leq N \leq 10^5$ * $0 \leq M \leq 10^5$ * $1 \leq A_i,B_i \leq N$ * $0 \leq C_i,D_i \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $A_1$ $B_1$ $C_1$ $D_1$ $\vdots$ $A_M$ $B_M$ $C_M$ $D_M$ [samples]

Samples

Input #1

2 1
1 2 2 3

Output #1

4

We will first stay in City $1$ until time $1$. Then, at time $1$, we will start going through Road $1$, which will take $2+\left\lfloor \frac{3}{1+1} \right\rfloor = 3$ time units before reaching City $2$ at time $4$.
It is impossible to reach City $2$ earlier than time $4$.

Input #2

Output #2

3

There may be multiple roads connecting the same pair of cities, and a road going from a city to the same city.

Input #3

4 2
1 2 3 4
3 4 5 6

Output #3

\-1

There may be no path from City $1$ to City $N$.

Input #4

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Rush Hour 2",
    "description": {
      "content": "The Republic of AtCoder has $N$ cities and $M$ roads. The cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ connects City $A_i$ and City $B_i$ bidirectionally. ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc204_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "The Republic of AtCoder has $N$ cities and $M$ roads.\nThe cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ connects City $A_i$ and City $B_i$ bidirectionally.\n...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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