Problem p04052 - Generation 1

Orig Description

Score : 2000 points
Problem StatementYou are given a permutation P_1 ... P_N of the set {1, 2, ..., N}.
You can apply the following operation to this permutation, any number of times (possibly zero):
Choose two indices i,j (1 ≦ i < j ≦ N), such that j - i ≧ K and |P_i - P_j| = 1.
Then, swap the values of P_i and P_j.
Among all permutations that can be obtained by applying this operation to the given permutation, find the lexicographically smallest one.
Constraints
2≦N≦500,000
1≦K≦N-1
P is a permutation of the set {1, 2, ..., N}.
InputThe input is given from Standard Input in the following format:
N K
P_1 P_2 ... P_N
OutputPrint the lexicographically smallest permutation that can be obtained.
Sample Input 14 2
4 2 3 1
Sample Output 12
1
4
3
One possible way to obtain the lexicographically smallest permutation is shown below:
4 2 3 1
4 1 3 2
3 1 4 2
2 1 4 3
Sample Input 25 1
5 4 3 2 1
Sample Output 21
2
3
4
5
Sample Input 38 3
4 5 7 8 3 1 2 6
Sample Output 31
2
6
7
5
3
4
8

Extracted Specification

An integer N (2 ≤ N ≤ 500,000), representing some quantity or size.
An integer K (1 ≤ K ≤ N-1), representing another quantity.
A list of integers P of size N, which is a permutation of numbers from 1 to N.

### Example Input:

```
4 2
4 2 3 1
```

### Function Signature:

Write a function f(N, K, P) that takes in the input.

```python
def f(N: int, K: int, P: List[int]):
    '''
    N: an integer
    K: an integer
    P: a list of integers
    '''
```

Test Case Generator

Sure! Let's create a test case generator for the given problem. The generator will produce random values for \(N\), \(K\), and the permutation \(P\) based on the problem constraints.

Here's the code for the test case generator:

```python
import random

def generate_permutation(N):
    P = list(range(1, N + 1))
    random.shuffle(P)
    return P

def tcgen():
    N = random.randint(2, 500000)
    K = random.randint(1, N - 1)
    P = generate_permutation(N)
    
    return N, K, P

def main():
    N, K, P = tcgen()
    print(N, K)
    print(' '.join(map(str, P)))

if __name__ == "__main__":
    main()
```

### Explanation:
1. **generate_permutation(N)**: This function generates a random permutation of the numbers from 1 to \(N\).
2. **tcgen()**:
    - It randomly selects a value for \(N\) within the range [2, 500000].
    - It randomly selects a value for \(K\) within the range [1, \(N-1\)].
    - It generates a random permutation \(P\) of size \(N\).
3. **main()**:
    - Calls the `tcgen()` function to get a random test case.
    - Prints the values of \(N\), \(K\), and the permutation \(P\) in the required format.

You can run this script to generate random test cases for the given problem. Each time you run it, it will produce a new test case. Here are a few example outputs:

### Example Output:
```
4 2
4 3 2 1
```

```
5 1
5 1 3 2 4
```

```
8 3
4 7 8 5 1 6 3 2
```

This way, you can generate an extensive set of test cases to test the robustness and correctness of your solution to the problem.

Extract Arguments

def extract_arguments(fh):
    N, K = map(int, fh.readline().strip().split())
    P = list(map(int, fh.readline().strip().split()))
    return N, K, P

# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh:
#         N, K, P = extract_arguments(fh)
#         f(N, K, P)