Train Service Planning

AtCoder

IDagc011_f

Time2000ms

Memory256MB

Difficulty—

There is a railroad in Takahashi Kingdom. The railroad consists of $N$ sections, numbered $1$, $2$, ..., $N$, and $N+1$ stations, numbered $0$, $1$, ..., $N$. Section $i$ directly connects the stations $i-1$ and $i$. A train takes exactly $A_i$ minutes to run through section $i$, regardless of direction. Each of the $N$ sections is either single-tracked over the whole length, or double-tracked over the whole length. If $B_i = 1$, section $i$ is single-tracked; if $B_i = 2$, section $i$ is double-tracked. Two trains running in opposite directions can cross each other on a double-tracked section, but not on a single-tracked section. Trains can also cross each other at a station. Snuke is creating the timetable for this railroad. In this timetable, the trains on the railroad run every $K$ minutes, as shown in the following figure. Here, bold lines represent the positions of trains running on the railroad. (See Sample 1 for clarification.) ![image](https://atcoder.jp/img/agc011/a5c221ce77ab6ee8aee48e75a4e5c969.png) When creating such a timetable, find the minimum sum of the amount of time required for a train to depart station $0$ and reach station $N$, and the amount of time required for a train to depart station $N$ and reach station $0$. It can be proved that, if there exists a timetable satisfying the conditions in this problem, this minimum sum is always an integer. Formally, the times at which trains arrive and depart must satisfy the following: * Each train either departs station $0$ and is bound for station $N$, or departs station $N$ and is bound for station $0$. * Each train takes exactly $A_i$ minutes to run through section $i$. For example, if a train bound for station $N$ departs station $i-1$ at time $t$, the train arrives at station $i$ exactly at time $t+A_i$. * Assume that a train bound for station $N$ arrives at a station at time $s$, and departs the station at time $t$. Then, the next train bound for station $N$ arrives at the station at time $s+K$, and departs the station at time $t+K$. Additionally, the previous train bound for station $N$ arrives at the station at time $s-K$, and departs the station at time $t-K$. This must also be true for trains bound for station $0$. * Trains running in opposite directions must not be running on the same single-tracked section (except the stations at both ends) at the same time. ## Constraints * $1 \leq N \leq 100000$ * $1 \leq K \leq 10^9$ * $1 \leq A_i \leq 10^9$ * $A_i$ is an integer. * $B_i$ is either $1$ or $2$. ## Input The input is given from Standard Input in the following format: $N$ $K$ $A_1$ $B_1$ $A_2$ $B_2$ : $A_N$ $B_N$ ## Partial Score * In the test set worth $500$ points, all the sections are single-tracked. That is, $B_i = 1$. * In the test set worth another $500$ points, $N \leq 200$. [samples]

Samples

Input #1

Output #1

26

For example, the sum of the amount of time in question will be $26$ minutes in the following timetable:
![image](https://atcoder.jp/img/agc011/a5c221ce77ab6ee8aee48e75a4e5c969.png)
In this timetable, the train represented by the red line departs station $0$ at time $0$, arrives at station $1$ at time $4$, departs station $1$ at time $5$, arrives at station $2$ at time $8$, and so on.

Input #2

1 10
10 1

Output #2

\-1

Input #3

Output #3

Input #4

20 987654321
129662684 2
162021979 1
458437539 1
319670097 2
202863355 1
112218745 1
348732033 1
323036578 1
382398703 1
55854389 1
283445191 1
151300613 1
693338042 2
191178308 2
386707193 1
204580036 1
335134457 1
122253639 1
824646518 2
902554792 2

Output #4

14829091348

API Response (JSON)

{
  "problem": {
    "name": "Train Service Planning",
    "description": {
      "content": "There is a railroad in Takahashi Kingdom. The railroad consists of $N$ sections, numbered $1$, $2$, ..., $N$, and $N+1$ stations, numbered $0$, $1$, ..., $N$. Section $i$ directly connects the station",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc011_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a railroad in Takahashi Kingdom. The railroad consists of $N$ sections, numbered $1$, $2$, ..., $N$, and $N+1$ stations, numbered $0$, $1$, ..., $N$. Section $i$ directly connects the station...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments