Fuel

AtCoder

IDawtf2024_c

Time2000ms

Memory256MB

Difficulty—

Solve the following problem for $T$ test cases. You are currently at coordinate $0$ on a number line, and you want to reach coordinate $L$. You will use a car for the movement. This car runs on two types of fuel: types $1$ and $2$. For each type, the car has a fuel tank of that type with a capacity of $C$ liters. Currently, both fuel tanks are full. The car can consume $x$ liters of either type of fuel (where $x$ is any positive integer not exceeding the remaining fuel) to move $x$ units in any direction on the number line. You can choose which type of fuel to use for each movement. There are $N$ fuel stations on the number line. The $i$\-th fuel station is located at coordinate $X_i$. When the car is exactly at coordinate $X_i$, you can buy type-$K_i$ fuel at a cost of $1$ per liter. Of course, you cannot buy more fuel than allowed by the tank capacity. Determine whether it is possible to reach coordinate $L$. If it is possible, find the minimum total cost of purchasing fuel. ## Constraints * $1 \leq T \leq 250000$ * $1 \leq N \leq 5000$ * $1 \leq L \leq 10^9$ * $1 \leq C \leq 10^9$ * $0 < X_1 < X_2 < \cdots < X_N < L$ * $K_i=1$ or $2$ * The sum of $N^2$ across all test cases in a single input does not exceed $5000^2$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $case_1$ $case_2$ $\vdots$ $case_T$ Each test case is given in the following format: $N$ $L$ $C$ $X_1$ $X_2$ $\cdots$ $X_N$ $K_1$ $K_2$ $\cdots$ $K_N$ [samples]

Samples

Input #1

5
1 10 4
7
1
1 10 6
7
1
2 12 3
5 7
1 1
2 12 3
5 7
1 2
20 749013197 23809523
46981984 70791437 118235723 132421762 180040807 203849360 251468335 275277857 322889975 346699150 394318091 418113855 465732891 489532137 537144103 558852533 606466719 630275002 677584754 701394209
1 2 2 1 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2

Output #1

2
0
-1
6
585545066743659

Let us explain the first test case. You can reach coordinate $L$ with a cost of $2$ as follows:

*   Consume $3$ liters of type-$1$ fuel to move from coordinate $0$ to $3$. The remaining type-$1$ fuel is $1$ liter.
*   Consume $4$ liters of type-$2$ fuel to move from coordinate $3$ to $7$. The remaining type-$2$ fuel is $0$ liters.
*   Buy $2$ liters of type-$1$ fuel at the fuel station. The remaining type-$1$ fuel is $3$ liters.
*   Consume $3$ liters of type-$1$ fuel to move from coordinate $7$ to $10$. The remaining type-$1$ fuel is $0$ liters.

It is impossible to reach coordinate $L$ with a cost less than $2$. Thus, the answer is $2$.

API Response (JSON)

{
  "problem": {
    "name": "Fuel",
    "description": {
      "content": "Solve the following problem for $T$ test cases. You are currently at coordinate $0$ on a number line, and you want to reach coordinate $L$. You will use a car for the movement. This car runs on two ty",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "awtf2024_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Solve the following problem for $T$ test cases.\nYou are currently at coordinate $0$ on a number line, and you want to reach coordinate $L$.\nYou will use a car for the movement. This car runs on two ty...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments