Problem p02661 - Generation 2

Orig Description

Score : 500 points
Problem StatementThere are N integers X_1, X_2, \cdots, X_N, and we know that A_i \leq X_i \leq B_i.
Find the number of different values that the median of X_1, X_2, \cdots, X_N can take.
NotesThe median of X_1, X_2, \cdots, X_N is defined as follows. Let x_1, x_2, \cdots, x_N be the result of sorting X_1, X_2, \cdots, X_N in ascending order.
If N is odd, the median is x_{(N+1)/2};
if N is even, the median is (x_{N/2} + x_{N/2+1}) / 2.
Constraints
2 \leq N \leq 2 \times 10^5
1 \leq A_i \leq B_i \leq 10^9
All values in input are integers.
InputInput is given from Standard Input in the following format:
N
A_1 B_1
A_2 B_2
:
A_N B_N
OutputPrint the answer.
Sample Input 12
1 2
2 3
Sample Output 13
If X_1 = 1 and X_2 = 2, the median is \frac{3}{2};
if X_1 = 1 and X_2 = 3, the median is 2;
if X_1 = 2 and X_2 = 2, the median is 2;
if X_1 = 2 and X_2 = 3, the median is \frac{5}{2}.
Thus, the median can take three values: \frac{3}{2}, 2, and \frac{5}{2}.
Sample Input 23
100 100
10 10000
1 1000000000
Sample Output 29991

Extracted Specification

    
An integer N (2 ≤ N ≤ 2 * 10^5), representing some quantity or size.
N pairs of integers (A_i, B_i) where 1 ≤ A_i ≤ B_i ≤ 10^9.

### Example Input:

```
3
100 100
10 10000
1 1000000000
```

### Function Signature:
Write a function f(N, AB) that takes in the input.
def f(N: int, AB: List[Tuple[int, int]]): 
    ''' 
    N: an integer 
    AB: a list of tuples, where each tuple contains two integers 
    '''

Test Case Generator

Sure! Here is a test case generator for the given competitive programming problem:

```python
import random

def tcgen():
    # Generate a random N within the given constraints
    N = random.randint(2, 2 * 10**5)
    
    # Generate the array of (A_i, B_i) pairs
    AB = []
    for _ in range(N):
        A = random.randint(1, 10**9)
        B = random.randint(A, 10**9)
        AB.append((A, B))
    
    return N, AB

# Example usage:
N, AB = tcgen()
print(N)
for a, b in AB:
    print(a, b)
```

Explanation of the generator:

1. **N generation**: It generates an integer \(N\) such that \(2 \leq N \leq 2 \times 10^5\).
2. **(A_i, B_i) pairs generation**: It generates \(N\) pairs of integers (A_i, B_i) such that \(1 \leq A_i \leq B_i \leq 10^9\). Each A_i is a random integer, and B_i is a random integer that is at least A_i to ensure \(A_i \leq B_i\).

Here is the test case generator in action:

```python
# Example usage
if __name__ == "__main__":
    N, AB = tcgen()
    print(N)
    for a, b in AB:
        print(a, b)
```

This will print a valid test case for the problem. Each run will generate a different test case due to the randomness.

Extract Arguments

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