Here's an analysis of the Goal Parser problem and a solution:

Naive Solution

A straightforward approach is to iterate through the input string command and check for the specific patterns "G", "()", and "(al)". When one of these patterns is found, append the corresponding interpretation to the result string.

Code (Python):

def interpret_naive(command: str) -> str:
    result = ""
    i = 0
    while i < len(command):
        if command[i] == 'G':
            result += 'G'
            i += 1
        elif command[i:i+2] == '()':
            result += 'o'
            i += 2
        elif command[i:i+4] == '(al)':
            result += 'al'
            i += 4
        else:
            i += 1 # Handle unexpected characters (optional)
    return result

Optimal Solution

The optimal solution also involves iterating through the string, but uses string replacement methods for efficiency.

Code (Python):

def interpret_optimal(command: str) -> str:
    command = command.replace('()', 'o')
    command = command.replace('(al)', 'al')
    return command

Big O Analysis

• Time Complexity: O(n), where n is the length of the input string command. Both the naive and optimal solutions iterate through the string once.
• Space Complexity: O(1). The space used is constant because the replacements are done in place, and we're only storing a few extra variables.

Edge Cases

• Empty String: If the input string is empty, the function should return an empty string. The provided solutions handle this case correctly.
• Invalid Input: If the input contains characters other than "G", "()", and "(al)", the naive solution (with the else condition included) would skip over them, while the optimal solution could produce unexpected results. The problem statement does not specify how to handle invalid input, so we assume it will not occur.

Summary

The optimal solution using string replacement is generally more concise and efficient than the naive iterative approach, though both have the same time complexity in Big O notation. The choice depends on the specific requirements of the context in which the code is used.