Problem p03020 - Generation 3

Orig Description

Score : 1200 points
Problem StatementSnuke is playing with red and blue balls, placing them on a two-dimensional plane.
First, he performed N operations to place red balls. In the i-th of these operations, he placed RC_i red balls at coordinates (RX_i,RY_i).
Then, he performed another N operations to place blue balls. In the i-th of these operations, he placed BC_i blue balls at coordinates (BX_i,BY_i).
The total number of red balls placed and the total number of blue balls placed are equal, that is, \sum_{i=1}^{N} RC_i = \sum_{i=1}^{N} BC_i. Let this value be S.
Snuke will now form S pairs of red and blue balls so that every ball belongs to exactly one pair.
Let us define the score of a pair of a red ball at coordinates (rx, ry) and a blue ball at coordinates (bx, by) as |rx-bx| + |ry-by|.
Snuke wants to maximize the sum of the scores of the pairs.
Help him by finding the maximum possible sum of the scores of the pairs.
Constraints
1 \leq N \leq 1000
0 \leq RX_i,RY_i,BX_i,BY_i \leq 10^9
1 \leq RC_i,BC_i \leq 10
\sum_{i=1}^{N} RC_i = \sum_{i=1}^{N} BC_i
All values in input are integers.
InputInput is given from Standard Input in the following format:
N
RX_1 RY_1 RC_1
RX_2 RY_2 RC_2
\vdots
RX_N RY_N RC_N
BX_1 BY_1 BC_1
BX_2 BY_2 BC_2
\vdots
BX_N BY_N BC_N
OutputPrint the maximum possible sum of the scores of the pairs.
Sample Input 12
0 0 1
3 2 1
2 2 1
5 0 1
Sample Output 18
If we pair the red ball at coordinates (0,0) and the blue ball at coordinates (2,2), the score of this pair is |0-2| + |0-2|=4.
Then, if we pair the red ball at coordinates (3,2) and the blue ball at coordinates (5,0), the score of this pair is |3-5| + |2-0|=4.
Making these two pairs results in the total score of 8, which is the maximum result.
Sample Input 23
0 0 1
2 2 1
0 0 2
1 1 1
1 1 1
3 3 2
Sample Output 216
Snuke may have performed multiple operations at the same coordinates.
Sample Input 310
582463373 690528069 8
621230322 318051944 4
356524296 974059503 6
372751381 111542460 9
392867214 581476334 6
606955458 513028121 5
882201596 791660614 9
250465517 91918758 3
618624774 406956634 6
426294747 736401096 5
974896051 888765942 5
726682138 336960821 3
715144179 82444709 6
599055841 501257806 6
390484433 962747856 4
912334580 219343832 8
570458984 648862300 6
638017635 572157978 10
435958984 585073520 7
445612658 234265014 6
Sample Output 345152033546

Extracted Specification

### Canonicalized Input Description:

An integer N (1 ≤ N ≤ 1000), representing some quantity or size.  
A list of N tuples, each containing two integers (0 ≤ integer ≤ 1e9) and another integer (1 ≤ integer ≤ 10).  
Another list of N tuples, each containing two integers (0 ≤ integer ≤ 1e9) and another integer (1 ≤ integer ≤ 10).  

### Example Input:

```
2
0 0 1
3 2 1
2 2 1
5 0 1
```

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

Test Case Generator

Sure, I can help you write a test case generator for the given problem statement. Here's how you can do it in Python:

```python
import random

def tcgen():
    # Generate a random N within the given constraint
    N = random.randint(1, 1000)
    
    # Initialize variables to keep track of the total number of red and blue balls
    total_red_balls = 0
    total_blue_balls = 0
    
    # Create lists to store the operations for red and blue balls
    red_operations = []
    blue_operations = []
    
    # Generate red ball operations
    for _ in range(N):
        RX = random.randint(0, 10**9)
        RY = random.randint(0, 10**9)
        RC = random.randint(1, 10)
        red_operations.append((RX, RY, RC))
        total_red_balls += RC
    
    # Generate blue ball operations, making sure the total number of blue balls equals total number of red balls
    for _ in range(N):
        BX = random.randint(0, 10**9)
        BY = random.randint(0, 10**9)
        
        # Ensure the sum of blue balls matches the sum of red balls
        if _ == N - 1:
            BC = total_red_balls - total_blue_balls
        else:
            BC = random.randint(1, min(10, total_red_balls - total_blue_balls - (N - _ - 1)))
        
        blue_operations.append((BX, BY, BC))
        total_blue_balls += BC
    
    # Format the input
    input_data = f"{N}\n"
    for op in red_operations:
        input_data += f"{op[0]} {op[1]} {op[2]}\n"
    for op in blue_operations:
        input_data += f"{op[0]} {op[1]} {op[2]}\n"
    
    return input_data

# Example usage:
print(tcgen())
```

### Explanation:
1. **N Generation**: We generate a random integer \( N \) between 1 and 1000.
2. **Red Ball Operations**:
    - For each operation, generate random coordinates \((RX, RY)\) within the range [0, 10^9].
    - Generate a random count \( RC \) within the range [1, 10].
    - Keep track of the total count of red balls.
3. **Blue Ball Operations**:
    - Similarly, generate random coordinates \((BX, BY)\) within the range [0, 10^9].
    - For the count \( BC \), ensure that the sum of all \( BC \) equals the sum of all \( RC \). To achieve this:
      - For each operation except the last, generate a random count \( BC \) such that the remaining counts can still fulfill the constraint.
      - For the last operation, set \( BC \) to ensure the sum of \( BC \) matches the sum of \( RC \).
4. **Formatting**: Combine the generated data into the required input format.

This generator will give you a variety of test cases within the constraints, ensuring the sums of red and blue balls are equal.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    red_balls = []
    blue_balls = []
    for _ in range(N):
        RX, RY, RC = map(int, fh.readline().strip().split())
        red_balls.append((RX, RY, RC))
    for _ in range(N):
        BX, BY, BC = map(int, fh.readline().strip().split())
        blue_balls.append((BX, BY, BC))
    return N, red_balls, blue_balls

# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh: 
#     N, red_balls, blue_balls = extract_arguments(fh)
#     f(N, red_balls, blue_balls)