Problem p02661 - Generation 3

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.  
A list of N tuples (A_i, B_i), where each tuple contains two integers:
- A_i (1 ≤ A_i ≤ 10^9), representing a starting value.
- B_i (A_i ≤ B_i ≤ 10^9), representing an ending value.

### Example Input:

```
2
1 2
2 3
```

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

Test Case Generator

To generate test cases for the problem "Find the number of different values that the median of X_1, X_2, \cdots, X_N can take", we need to account for various cases within the constraints provided. The constraints are:

- \(2 \leq N \leq 2 \times 10^5\)
- \(1 \leq A_i \leq B_i \leq 10^9\)

Here's a Python function to generate test cases for the problem:

```python
import random

def tcgen():
    # Generate a random value for N within the constraints
    N = random.randint(2, 2 * 10**5)
    
    # Generate pairs (A_i, B_i) where 1 <= A_i <= B_i <= 10^9
    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)
```

This function generates a random integer \(N\) within the specified range and then creates \(N\) pairs of integers \((A_i, B_i)\) such that \(1 \leq A_i \leq B_i \leq 10^9\). Each pair is generated to ensure \(A_i \leq B_i\).

You can use this function to generate test cases for the problem. Note that the output of the function is in the correct format for input to the competitive programming problem.

### Explanation:
1. **N Generation**: Selects a random value for \(N\) which determines the number of pairs.
2. **Pair Generation**: For each pair, selects \(A_i\) randomly between 1 and \(10^9\). Then, \(B_i\) is selected such that it is greater than or equal to \(A_i\).

This should allow you to test the problem for a wide range of inputs, ensuring robustness and correctness of your solution.

Extract Arguments

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