Problem p03111 - Generation 1

Orig Description

Score : 300 points
Problem StatementYou have N bamboos. The lengths (in centimeters) of these are l_1, l_2, ..., l_N, respectively.
Your objective is to use some of these bamboos (possibly all) to obtain three bamboos of length A, B, C. For that, you can use the following three kinds of magics any number:
Extension Magic: Consumes 1 MP (magic point). Choose one bamboo and increase its length by 1.
Shortening Magic: Consumes 1 MP. Choose one bamboo of length at least 2 and decrease its length by 1.
Composition Magic: Consumes 10 MP. Choose two bamboos and combine them into one bamboo. The length of this new bamboo is equal to the sum of the lengths of the two bamboos combined. (Afterwards, further magics can be used on this bamboo.)
At least how much MP is needed to achieve the objective?
Constraints
3 \leq N \leq 8
1 \leq C < B < A \leq 1000
1 \leq l_i \leq 1000
All values in input are integers.
InputInput is given from Standard Input in the following format:
N A B C
l_1
l_2
:
l_N
OutputPrint the minimum amount of MP needed to achieve the objective.
Sample Input 15 100 90 80
98
40
30
21
80
Sample Output 123
We are obtaining three bamboos of lengths 100, 90, 80 from five bamboos 98, 40, 30, 21, 80. We already have a bamboo of length 80, and we can obtain bamboos of lengths 100, 90 by using the magics as follows at the total cost of 23 MP, which is optimal.
Use Extension Magic twice on the bamboo of length 98 to obtain a bamboo of length 100. (MP consumed: 2)
Use Composition Magic on the bamboos of lengths 40, 30 to obtain a bamboo of length 70. (MP consumed: 10)
Use Shortening Magic once on the bamboo of length 21 to obtain a bamboo of length 20. (MP consumed: 1)
Use Composition Magic on the bamboo of length 70 obtained in step 2 and the bamboo of length 20 obtained in step 3 to obtain a bamboo of length 90. (MP consumed: 10)
Sample Input 28 100 90 80
100
100
90
90
90
80
80
80
Sample Output 20
If we already have all bamboos of the desired lengths, the amount of MP needed is 0. As seen here, we do not necessarily need to use all the bamboos.
Sample Input 38 1000 800 100
300
333
400
444
500
555
600
666
Sample Output 3243

Extracted Specification

Canonicalized Input Description:

An integer \(N\) (3 ≤ \(N\) ≤ 8), representing some quantity or size.
Three distinct integers \(A\), \(B\), and \(C\) (1 ≤ \(C\) < \(B\) < \(A\) ≤ 1000).
A list of \(N\) integers \(l\), where each integer \(l_i\) (1 ≤ \(l_i\) ≤ 1000).

### Example Input:

```
5 100 90 80
98
40
30
21
80
```

### Function Signature:
Write a function `f(N, (A, B, C), l)` that takes in the input.
```python
def f(N: int, targets: Tuple[int, int, int], lengths: List[int]):
    '''
    N: an integer
    targets: a tuple of three integers
    lengths: a list of integers
    '''
```

Test Case Generator

Sure! Below is a Python test case generator for the given bamboo problem.

```python
import random

def tcgen():
    # Constraints
    N = random.randint(3, 8)
    A = random.randint(1, 1000)
    B = random.randint(1, A - 1)
    C = random.randint(1, B - 1)
    
    # Ensure l_i values are within the given constraints
    lengths = [random.randint(1, 1000) for _ in range(N)]
    
    return N, A, B, C, lengths

def format_test_case(N, A, B, C, lengths):
    test_case = f"{N} {A} {B} {C}\n" + "\n".join(map(str, lengths))
    return test_case

# Example usage
for _ in range(5):  # Generate 5 random test cases
    N, A, B, C, lengths = tcgen()
    print(format_test_case(N, A, B, C, lengths))
    print()
```

This function `tcgen` generates test cases for the problem, ensuring that all constraints are met. It then formats the output to match the expected input format for the problem. You can adjust the number of test cases generated by changing the range in the example usage.

Let's break down what the generator does:
1. **N:** Randomly selects the number of bamboos between 3 and 8.
2. **A, B, C:** Randomly selects the lengths of the desired bamboos ensuring \(C < B < A \leq 1000\).
3. **lengths:** Generates a list of N random bamboo lengths, each between 1 and 1000.

The `format_test_case` function takes these values and formats them into the standard input format for the problem, which is then printed out.

Feel free to use and modify this generator as needed for your testing purposes!

Extract Arguments

def extract_arguments(fh):
    N, A, B, C = map(int, fh.readline().strip().split())
    lengths = [int(fh.readline().strip()) for _ in range(N)]
    return N, A, B, C, lengths