Check whether two strings are equivalent or not according to given condition

Given two strings A and B of equal size. Two strings are equivalent either of the following conditions hold true: 
1) They both are equal. Or, 
2) If we divide the string A into two contiguous substrings of same size A₁ and A₂ and string B into two contiguous substrings of same size B₁ and B₂, then one of the following should be correct: 

• A₁ is recursively equivalent to B₁ and A₂ is recursively equivalent to B₂
• A₁ is recursively equivalent to B₂ and A₂ is recursively equivalent to B₁

Check whether given strings are equivalent or not. Print YES or NO.
Examples:

Input : A = "aaba", B = "abaa" 
Output : YES 
Explanation : Since condition 1 doesn't hold true, we can divide string A into "aaba" = "aa" + "ba" and string B into "abaa" = "ab" + "aa". Here, 2^nd subcondition holds true where A₁ is equal to B₂ and A₂ is recursively equal to B₁
Input : A = "aabb", B = "abab" 
Output : NO 

Naive Solution : A simple solution is to consider all possible scenarios. Check first if the both strings are equal return "YES", otherwise divide the strings and check if A₁ = B₁ and A₂ = B₂ or A₁ = B₂ and A₂ = B₁ by using four recursive calls. Complexity of this solution would be O(n²), where n is the size of the string.

Algorithm

is_recursive_equiv(A, B):
 if A == B:
 return "YES" // base case: A and B are already equivalent
 if length(A) is odd or length(B) is odd:
 return "NO" // base case: odd-length strings cannot be recursively equivalent
 n = length(A)
 A1 = substring(A, 0, n/2) // divide A into two halves
 A2 = substring(A, n/2)
 B1 = substring(B, 0, n/2) // divide B into two halves
 B2 = substring(B, n/2)
 if (is_recursive_equiv(A1, B1) == "YES" and is_recursive_equiv(A2, B2) == "YES") or
 (is_recursive_equiv(A1, B2) == "YES" and is_recursive_equiv(A2, B1) == "YES"):
 return "YES" // if corresponding halves are recursively equivalent, return "YES"
 return "NO" // otherwise, return "NO"

Implementation of above approach

YES
NO

Efficient Solution : Let's define following operation on string S. We can divide it into two halves and if we want we can swap them. And also, we can recursively apply this operation to both of its halves. By careful observation, we can see that if after applying the operation on some string A, we can obtain B, then after applying the operation on B we can obtain A. And for the given two strings, we can recursively find the least lexicographically string that can be obtained from them. Those obtained strings if are equal, answer is YES, otherwise NO. For example, least lexicographically string for "aaba" is "aaab". And least lexicographically string for "abaa" is also "aaab". Hence both of these are equivalent.

Steps to follow to implement the above approach:

• Define a function leastLexiString(s) that takes a string s as input and returns the least lexicographical string that can be formed       by rearranging the letters of s.
  • If the length of s is odd, return s as it is. This is the base case.
  • Divide the string s into two halves, x and y.
  • Recursively call leastLexiString() on x and y to find the least lexicographical strings that can be formed from these two                      halves.
  • return min(x+y,y+x).
• Define a function areEquivalent(a, b) that takes two strings a and b as input and returns true if the least lexicographical strings         that can be formed from a and b are equal, false otherwise.
  • Call leastLexiString() on a and b, and compare the results. If they are equal, return true, otherwise return false.

Below is the code to implement the above approach:

YES
NO

Time Complexity: O(N*logN), where N is the size of the string.
Auxiliary Space: O(logN)