Problem Description

The provided problem involves finding the largest string x that can divide two input strings str1 and str2. One string divides another if it can be repeated some number of times to obtain the other string. For example, if we have str1 = "abcabc" and str2 = "abc", then str2 divides str1 because we can concatenate str2 with itself twice to get str1. In other words, str1 is the result of appending str2 to itself multiple times. The challenge is to identify the largest common string that can divide both str1 and str2 in a similar fashion.

To illustrate further, if str1 is "ababab" and str2 is "abab", the string "ab" is the answer since it is the largest string that divides both str1 and str2.

Intuition

The underlying concept used in the provided solution is based on the mathematical idea of finding the greatest common divisor (GCD), but rather than numbers, we're dealing with strings. The key insight is that if there exists such a common string x that divides both str1 and str2, then the concatenation of str1 with str2 (str1 + str2) should be the same as the concatenation of str2 with str1 (str2 + str1). If this condition doesn't hold, it implies there is no such common string and the answer is an empty string.

Assuming the condition is true, we can find the length of the largest string x using the greatest common divisor of the lengths of str1 and str2. This is because the repeating pattern of string x must be a factor of both string lengths in order for it to divide both strings completely. Therefore, we find the GCD of the lengths of str1 and str2 and then return the substring of str1 up to that length. This substring is the largest common divider x.

Learn more about Math patterns.

Solution Approach

The solution provided takes advantage of Python's built-in gcd function from the [math](/problems/math-basics) library to calculate the greatest common divisor of the lengths of the two input strings str1 and str2. This is crucial for the solution, as the gcd represents the length of the largest string that can divide both strings, if such a string exists.

Here is how the solution unfolds:

1. Check for Compatibility: The first step in the provided solution checks whether the strings are compatible to have a common divisor by concatenating str1 with str2 and comparing it to str2 concatenated with str1. This is facilitated by the operation if str1 + str2 != str2 + str1, which ensures that the pattern of characters in both strings is compatible for them to have a common divisor string. If the check fails, the two strings cannot have a common divisor and the function immediately returns an empty string ''.

2. Find Length of the Common Divisor: If the strings pass the compatibility check, the next step is to determine the length of the largest common divisor string. This is where we use the gcd function, by calling n = gcd(len(str1), len(str2)). The gcd function takes the lengths of str1 and str2 and returns the greatest common divisor of these two numbers.

3. Extract the Common Divisor String: With the length n obtained from the gcd function, the final step is to extract the common divisor string from str1. This is accomplished by the expression return str1[:n]. The [:n] slice operation on str1 returns the substring of str1 from the beginning up to the n-th character (exclusive), which is the required largest string that divides both str1 and str2.

To summarize, the algorithm consists of concatenation for compatibility checking and the greatest common divisor calculation, both of which are simple, efficient operations. The lack of any complex data structures and the use of a mathematical pattern of common division make this solution both elegant and effective. It leverages the fact that if a common divisor string exists, the repeating pattern must align with the gcd of the string lengths.

Example Walkthrough

Let's use a small example to illustrate the solution approach. Suppose we have two strings str1 = "abab" and str2 = "ab". We need to find the largest string x that can divide both str1 and str2.

1. Check for Compatibility: We first check if concatenating str1 with str2 is equal to str2 concatenated with str1.

   • str1 + str2 = "abab" + "ab" = "ababab"
   • str2 + str1 = "ab" + "abab" = "ababab"

   Since str1 + str2 is equal to str2 + str1, the two strings are compatible, and therefore, it's possible they have a common divisor string.

2. Find Length of the Common Divisor: Next, we find the greatest common divisor of the lengths of str1 (4) and str2 (2).

   • gcd(4, 2) = 2

   The gcd tells us that the length of the largest string that possibly divides both str1 and str2 is 2.

3. Extract the Common Divisor String: Since the gcd is 2, we take the first 2 characters from str1 to form our largest divisor string x.

   • str1[:2] = "ab"

   So, "ab" is the string we expect to divide both str1 and str2. To verify, we can see that:

   • str1 divided by "ab" is "abab" / "ab" = "ab", which is the repetition of "ab" twice.
   • str2 divided by "ab" is "ab" / "ab" = "", which is the repetition of "ab" once.

The largest string that can divide both str1 and str2 is "ab", which matches our result from the solution approach. The solution is elegant and efficient, leveraging simple concatenation and the mathematical concept of greatest common divisor to determine the existence and length of a common divisor string.

Solution Implementation

Time and Space Complexity

The code contains the gcdOfStrings method that finds the greatest common divisor (GCD) of lengths of two strings str1 and str2 to determine the largest string that can be repeatedly used to construct str1 and str2.

Time Complexity:

The time complexity of this function mainly depends on two operations: the string concatenation operation (str1 + str2 and str2 + str1) and the greatest common divisor computation (gcd(len(str1), len(str2))).

1. String Concatenation: If str1 is of length n and str2 of length m, the concatenation will take O(n + m) time as each character from both strings need to be combined once.

2. Checking String Equality: Comparing the concatenated strings takes O(n + m) time. If the strings differ, the method returns early.

3. Greatest Common Divisor: The gcd function typically employs Euclid's algorithm, which has a time complexity of O(log(min(n, m))) where n and m are the lengths of the strings.

Therefore, the overall time complexity combines these operations resulting in O(n + m + log(min(n, m))). However, the dominating factor for large values of n and m will tend to be the concatenation and comparison, so we can approximate the time complexity as O(n + m).

Space Complexity:

The space complexity of the gcdOfStrings function can be analyzed as follows:

1. Temporary Strings: The creation of concatenated strings str1 + str2 and str2 + str1 requires additional space of O(n + m).

2. GCD Computation: The gcd function itself may use constant space, O(1), if the Euclidean algorithm is implemented in an iterative manner.

As such, no additional space that grows with the input size is required except for the string concatenation. Hence, the space complexity is O(n + m).

Learn more about how to find time and space complexity quickly using problem constraints.