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 tuples (A, B) of size N, where each integer in the tuple satisfies 1 ≤ A ≤ B ≤ 10^9.
### Example Input:
```
2
1 2
2 3
```
### Function Signature:
Write a function f(inputs) that takes in the input.
def f(inputs: List[Tuple[int, int]]):
'''
inputs: a list of tuples, where each tuple contains two integers
'''
Test Case Generator
Sure, I'll write a test case generator for the given problem statement. The problem involves generating constraints and values of \( A_i \) and \( B_i \) for \( N \) integers, where \( 2 \leq N \leq 2 \times 10^5 \) and \( 1 \leq A_i \leq B_i \leq 10^9 \).
Here's the test case generator in Python:
```python
import random
def tcgen():
# Generate a random N in the valid range
N = random.randint(2, 2 * 10**5)
# Generate A and B arrays with valid constraints
A = []
B = []
for _ in range(N):
a = random.randint(1, 10**9)
b = random.randint(a, 10**9)
A.append(a)
B.append(b)
# Create the formatted input as a string
input_str = f"{N}\n"
for a, b in zip(A, B):
input_str += f"{a} {b}\n"
return input_str
# Example usage
print(tcgen())
```
This test case generator will create large and valid test cases according to the given constraints. The function `tcgen()` generates a random integer \( N \) within the specified range, and then for each \( i \), it generates random values for \( A_i \) and \( B_i \) such that \( 1 \leq A_i \leq B_i \leq 10^9 \). It then formats these values into a string that matches the expected input format for the problem.
You can call this function to generate different test cases, and you can use the output to test your solution to the competitive programming problem.
Extract Arguments
def extract_arguments(fh):
N = int(fh.readline().strip())
intervals = []
for _ in range(N):
A, B = map(int, fh.readline().strip().split())
intervals.append((A, B))
return N, intervals