What is the Greatest Common Divisor of Strings?

The greatest common divisor (GCD) of strings is a topic that often comes up in technical interviews for software developers. It concerns finding the largest string that can evenly divide two or more given strings. This concept is linked to the basic idea of divisibility in mathematics and can lead to interesting algorithms and string manipulations in programming.

Understanding the Concept

In simple terms, the GCD of two strings s1 and s2 is the longest string x such that both s1 and s2 can be constructed by concatenating multiple copies of x.

For example, consider the strings:

• s1 = "ABCABCABC"
• s2 = "ABCABC"

The GCD of these strings is "ABC", because both strings can be made by repeating the string "ABC" multiple times.

How to Find the GCD of Strings

Algorithm Overview

1. Check if the concatenation of s1 and s2 in both orders are the same:
   • If s1 + s2 is not equal to s2 + s1, then return an empty string as there is no common divisor.
2. If they are equal, compute the GCD of their lengths.
3. Use the length to slice one of the strings to obtain the GCD string.

Pseudocode

PlaintextCopy code
function gcdOfStrings(s1, s2):
    if (s1 + s2 != s2 + s1):
        return ""
    gcdLength = gcd(length(s1), length(s2))
    return s1.substring(0, gcdLength)

Example

Let’s see an example with code in Python:

PythonCopy code
def gcdOfStrings(s1: str, s2: str) -> str:
    if s1 + s2 != s2 + s1:
        return ""
    gcd_length = gcd(len(s1), len(s2))
    return s1[:gcd_length]

def gcd(x, y):
    while y:
        x, y = y, x % y
    return x

# Example usage
s1 = "ABCABCABC"
s2 = "ABCABC"
print(gcdOfStrings(s1, s2))  # Output: "ABC"

Example Interview Questions

1. Question: "How would you find the GCD of the strings 'ABABAB' and 'ABAB'?"

   Answer: First, concatenate the two strings in both possible orders:

   • 'ABABAB' + 'ABAB' = 'ABABABABAB'
   • 'ABAB' + 'ABABAB' = 'ABABABABAB' Since they are equal, we move forward. Next, compute the lengths:
   • Length of 'ABABAB' is 6
   • Length of 'ABAB' is 4

   The GCD of 6 and 4 is 2. Taking the substring of length 2 from either string gives us 'AB'. The GCD of the strings is 'AB'.

2. Question: "What will be the GCD of the strings 'XYZ' and 'XYZXYZ'?"

   Answer: Concatenating in both orders will give:

   • 'XYZ' + 'XYZXYZ' = 'XYZXYZXYZ'
   • 'XYZXYZ' + 'XYZ' = 'XYZXYZXYZ'

   Since they match, we proceed to find the lengths, where ‘XYZ’ is 3 and ‘XYZXYZ’ is 6. The GCD of 3 and 6 is 3. Taking the first 3 characters from either string gives us 'XYZ'. Therefore, the answer is 'XYZ'.

3. Question: "If given strings 'AA' and 'AAB', what will be their GCD?"

   Answer: Concatenating:

   • 'AA' + 'AAB' = 'AAAAB'
   • 'AAB' + 'AA' = 'AABAA'

   They do not match, so the GCD is an empty string ''. This means there is no common divisor for these strings.

Key Points to Remember

When preparing for interviews, focus on understanding how to manipulate strings and use algorithms efficiently. Make sure to practice writing clear and concise code, as demonstrating your problem-solving approach can often be more important than arriving at the correct answer immediately.