Problem p03952 - Generation 3

Orig Description

Score : 400 points
Problem StatementWe have a pyramid with N steps, built with blocks.
The steps are numbered 1 through N from top to bottom.
For each 1≤i≤N, step i consists of 2i-1 blocks aligned horizontally.
The pyramid is built so that the blocks at the centers of the steps are aligned vertically.
A pyramid with N=4 steps
Snuke wrote a permutation of (1, 2, ..., 2N-1) into the blocks of step N.
Then, he wrote integers into all remaining blocks, under the following rule:
The integer written into a block b must be equal to the median of the three integers written into the three blocks directly under b, or to the lower left or lower right of b.
Writing integers into the blocks
Afterwards, he erased all integers written into the blocks.
Now, he only remembers that the integer written into the block of step 1 was x.
Construct a permutation of (1, 2, ..., 2N-1) that could have been written into the blocks of step N, or declare that Snuke's memory is incorrect and such a permutation does not exist.
Constraints
2≤N≤10^5
1≤x≤2N-1
InputThe input is given from Standard Input in the following format:
N x
OutputIf no permutation of (1, 2, ..., 2N-1) could have been written into the blocks of step N, print No.
Otherwise, print Yes in the first line, then print 2N-1 lines in addition.
The i-th of these 2N-1 lines should contain the i-th element of a possible permutation.
Sample Input 14 4
Sample Output 1Yes
1
6
3
7
4
5
2
This case corresponds to the figure in the problem statement.
Sample Input 22 1
Sample Output 2No
No matter what permutation was written into the blocks of step N, the integer written into the block of step 1 would be 2.

Extracted Specification

An integer N (2 ≤ N ≤ 10^5), representing some quantity or size.
An integer x (1 ≤ x ≤ 2N-1).

### Example Input:

```
4 4
```

### Function Signature:
Write a function f(N, x) that takes in the input.
def f(N: int, x: int):
    '''
    N: an integer
    x: an integer
    '''

Test Case Generator

Sure, I can write a test case generator for the given problem. Here's the Python code for generating test cases:

```python
import random

def tcgen():
    N = random.randint(2, 10**5)
    x = random.randint(1, 2 * N - 1)
    return (N, x)

def main():
    for _ in range(10):  # Generating 10 test cases for example
        N, x = tcgen()
        print(N, x)

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

### Explanation:

1. **Random Generation of \( N \) and \( x \)**:
    - \( N \) is randomly chosen between 2 and \( 10^5 \) (inclusive) as per the problem constraints.
    - \( x \) is randomly chosen between 1 and \( 2N - 1 \) (inclusive) as per the problem constraints.
  
2. **Printing the Test Cases**:
    - The function `main()` is used to generate and print several test cases (in this case, 10 test cases).

### Example Output:

Running the `main()` function might produce output such as:

```
18904 25027
73290 146579
55660 111319
36885 73742
10234 20457
...
```

This output corresponds to different test cases of \( N \) and \( x \) that can be used to test your solution to the problem.

Feel free to adjust the number of test cases generated or the range of values if necessary.

Extract Arguments

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