Problem Analysis

Given a string containing only digits within parentheses, the task is to generate all valid coordinate pairs that could have produced the original string when punctuation was removed. The coordinates must adhere to specific formatting rules: no leading or trailing zeros in decimal components, and decimal points must be preceded by at least one digit.

Solution Approach

The solution employs a systematic enumeration strategy. First, the input string is stripped of parentheses. Then, all possible split points for the comma separator are considered. For each split, valid number representatinos are generated for both the left and right segments.

Algorithm Implementation

def generate_coordinates(s: str) -> List[str]:
    def generate_numbers(start: int, end: int) -> List[str]:
        results = []
        if start == end or s[start] != '0':
            results.append(s[start:end+1])
        
        for decimal_pos in range(start, end):
            integer_part = s[start:decimal_pos+1]
            fractional_part = s[decimal_pos+1:end+1]
            
            if len(integer_part) > 1 and integer_part[0] == '0':
                continue
            if fractional_part[-1] == '0':
                continue
                
            results.append(f'{integer_part}.{fractional_part}')
        
        return results
    
    s = s[1:-1]  # Remove parentheses
    n = len(s)
    coordinate_pairs = []
    
    for split_point in range(n - 1):
        left_options = generate_numbers(0, split_point)
        right_options = generate_numbers(split_point + 1, n - 1)
        
        for left_val in left_options:
            for right_val in right_options:
                coordinate_pairs.append(f'({left_val}, {right_val})')
    
    return coordinate_pairs

Key Validation Rules

• Integer Validation: If the entire segment represents an integer, it must not have leading zeros unless it's exactly "0"
• Decimal Validation: The integer part before the decimal point must not have leading zeros
• Fracsional Validation: The fractional part after the decimal point must not end with zero

Complexity Aanlysis

• Time Complexity: O(n³) where n is the string length
• Space Complexity: O(n³) for storing all possible coordinate combinations