algoadvance

You are given an initially empty string s. You are supposed to perform exactly n operations to obtain a string that contains only the letter ‘A’. Initially, you can only perform the following operations:

1. Copy All: You can copy all the characters present in the notepad (initially empty) and store it in the clipboard.
2. Paste: You can paste the characters present in the clipboard on the notepad.

You need to return the minimum number of operations required to get exactly n ‘A’ characters on the notepad.

Clarifying Questions

1. What is the initial state of the notepad and clipboard?
   • The notepad starts empty and the clipboard starts empty.
2. What counts as an operation?
   • Either a “Copy All” or a “Paste” operation counts as one step.
3. Is n guaranteed to be a positive integer?
   • Yes, n is always a positive integer.
4. Are there any constraints on n?
   • Typically, n would be a moderate value for an interview problem.

Strategy

To minimize the operations needed, we need to find the optimal sequence of Copy and Paste operations. One approach is to recognize that if we are dealing with a composite number of copies:

• We can break down the task of achieving n ‘A’s by finding the smallest factors.
• For each factor, we multiply the minimal steps needed to achieve the factor subproblem and add this to our running total of operations.

For a given n, if we can decompose it into smaller steps using its factors, we can iteratively compute the operations required for each factor until we achieve exactly n ‘A’s.

Code

Here’s the Python code to implement this strategy:

def minSteps(n: int) -> int:
    if n == 1:
        return 0
    
    def min_operations_to_get(factor):
        steps = 0
        divisor = 2
        while factor > 1:
            while factor % divisor == 0:
                steps += divisor
                factor //= divisor
            divisor += 1
        return steps

    return min_operations_to_get(n)

# Example usage
print(minSteps(3)) # returns 3
print(minSteps(7)) # returns 7
print(minSteps(12)) # returns 7 (2 copy+paste for 4, plus 3 copy+paste for 3)

Time Complexity

• The time complexity is decent for reasonable values of n, specifically it involves the factorization process which is O(n log log n) in practice due to the divisor increment and modulo operations. For most practical values of n, this is efficient.

The algorithm leverages prime factorization to find the minimum steps by breaking the problem into smaller sub-problems of finding the steps required to create factors of n.

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