Check if a string is a scrambled form of another string

Given two strings S1 and S2 of equal length, the task is to determine if S2 is a scrambled form of S1.

Scrambled string:
Given string str, we can represent it as a binary tree by partitioning it to two non-empty substrings recursively.

Note: Srambled string is not same as an Anagram

Below is one possible representation of str = “coder”:

    coder
   /    \
  co    der
 / \    /  \
c   o  d   er
           / \
          e   r

To scramble the string, we may choose any non-leaf node and swap its two children.
Suppose, we choose the node "co" and swap its two children, it produces a scrambled string "ocder".



    ocder
   /    \
  oc    der
 / \    /  \
o   c  d   er
           / \
          e   r

Thus, “ocder” is a scrambled string of “coder”.

Similarly, if we continue to swap the children of nodes “der” and “er”, it produces a scrambled string “ocred”.

    ocred
   /    \
  oc    red
 / \    /  \
o   c  re  d
       / \
      r   e

Thus, “ocred” is a scrambled string of “coder”.

Examples:

Input: S1=”coder”, S2=”ocder”
Output: Yes
Explanation:
“ocder” is a scrambled form of “coder”

Input: S1=”abcde”, S2=”caebd”
Output: No
Explanation:
“caebd” is not a scrambled form of “abcde”

Approach
In order to solve this problem, we are using Divide and Conquer approach.
Given two strings of equal length (say n+1), S1[0...n] and S2[0...n]. If S2 is a scrambled form of S1, then there must exist an index i such that at least one of the following conditions is true:

  • S2[0...i] is a scrambled string of S1[0...i] and S2[i+1...n] is a scrambled string of S1[i+1...n].
  • S2[0...i] is a scrambled string of S1[n-i...n] and S2[i+1...n] is a scrambled string of S1[0...n-i-1].

Note: An optimization step to consider here is to check beforehand if the two strings are anagrams of each other. If not, it indicates that the strings contain different characters and can’t be a scrambled form of each other.

Below is the implementation of the above approach:

CPP

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// C++ Program to check if a
// given string is a scrambled
// form of another string
  
#include <bits/stdc++.h>
using namespace std;
  
bool isScramble(string S1, string S2)
{
    // Strings of non-equal length
    // cant' be scramble strings
    if (S1.length() != S2.length()) {
        return false;
    }
  
    int n = S1.length();
  
    // Empty strings are scramble strings
    if (n == 0) {
        return true;
    }
  
    // Equal strings are scramble strings
    if (S1 == S2) {
        return true;
    }
  
    // Check for the condition of anagram
    string copy_S1 = S1, copy_S2 = S2;
  
    sort(copy_S1.begin(), copy_S1.end());
    sort(copy_S2.begin(), copy_S2.end());
  
    if (copy_S1 != copy_S2) {
        return false;
    }
  
    for (int i = 1; i < n; i++) {
  
        // Check if S2[0...i] is a scrambled
        // string of S1[0...i] and if S2[i+1...n]
        // is a scrambled string of S1[i+1...n]
        if (isScramble(S1.substr(0, i), S2.substr(0, i))
            && isScramble(S1.substr(i, n - i),
                          S2.substr(i, n - i))) {
            return true;
        }
  
        // Check if S2[0...i] is a scrambled
        // string of S1[n-i...n] and S2[i+1...n]
        // is a scramble string of S1[0...n-i-1]
        if (isScramble(S1.substr(0, i),
                       S2.substr(n - i, i))
            && isScramble(S1.substr(i, n - i),
                          S2.substr(0, n - i))) {
            return true;
        }
    }
  
    // If none of the above
    // conditions are satisfied
    return false;
}
  
// Driver Code
int main()
{
    string S1 = "coder";
    string S2 = "ocred";
  
    if (isScramble(S1, S2)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
  
    return 0;
}

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Output:

Yes

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