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
Canonicalized Input Description:
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
Certainly! Creating a test case generator for the given problem involves generating valid inputs that adhere to the constraints specified. Here's a Python function that generates such test cases:
```python
import random
def tcgen():
# Step 1: Generate a random N within the given constraints
N = random.randint(2, 10**5)
# Step 2: Generate a random x within the range 1 to 2N-1
x = random.randint(1, 2 * N - 1)
return N, x
# Function to generate multiple test cases
def generate_test_cases(num_cases):
test_cases = []
for _ in range(num_cases):
test_cases.append(tcgen())
return test_cases
# Example of generating 5 test cases
if __name__ == "__main__":
test_cases = generate_test_cases(5)
for case in test_cases:
print(f"Input: {case[0]} {case[1]}")
```
### Explanation:
1. **Generate a Random N**:
- `N` is chosen randomly within the constraints \(2 \leq N \leq 10^5\).
2. **Generate a Random x**:
- `x` is chosen randomly within the range \(1 \leq x \leq 2N-1\).
3. **Generate Multiple Test Cases**:
- A helper function `generate_test_cases` is defined to generate a specified number of test cases. This function calls `tcgen` multiple times to generate the desired number of test cases.
4. **Print Example Test Cases**:
- In the main block, we generate and print 5 example test cases.
This script will generate valid input pairs `(N, x)` adhering to the problem constraints, which can be used for testing the competitive programming solution.
Extract Arguments
def extract_arguments(fh):
N, x = map(int, fh.readline().strip().split())
return N, x