Scramble String LeetCode Solution

Difficulty Level Hard
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Problem Statement

Scramble String LeetCode Solution – We can scramble a string s to get a string t using the following algorithm:

  1. If the length of the string is 1, stop.
  2. If the length of the string is > 1, do the following:
    • Split the string into two non-empty substrings at a random index, i.e., if the string is s, divide it to x and y where s = x + y.
    • Randomly decide to swap the two substrings or to keep them in the same order. i.e., after this step, s may become s = x + y or s = y + x.
    • Apply step 1 recursively on each of the two substrings x and y.

Given two strings s1 and s2 of the same length, return true if s2 is a scrambled string of s1, otherwise, return false.

Input: s1 = "great", s2 = "rgeat"
Output: true
Explanation: One possible scenario applied on s1 is:
"great" --> "gr/eat" // divide at random index.
"gr/eat" --> "gr/eat" // random decision is not to swap the two substrings and keep them in order.
"gr/eat" --> "g/r / e/at" // apply the same algorithm recursively on both substrings. divide at random index each of them.
"g/r / e/at" --> "r/g / e/at" // random decision was to swap the first substring and to keep the second substring in the same order.
"r/g / e/at" --> "r/g / e/ a/t" // again apply the algorithm recursively, divide "at" to "a/t".
"r/g / e/ a/t" --> "r/g / e/ a/t" // random decision is to keep both substrings in the same order.
The algorithm stops now, and the result string is "rgeat" which is s2.
As one possible scenario led s1 to be scrambled to s2, we return true.

Explanation

The basic idea is to divide s1(s2) into two substrings with length k and len-k and check if the two substrings s1[0..k-1] and s1[k, len-1] are the scrambles of s2[0..k-1] and s2[k,len-1] or s2[len-k, len-1] and s2[0..len-k-1] via recursion. The straightforward recursion will be very slow due to many repeated recursive function calls. To speed up the recursion, we can use an unordered_map isScramblePair to save intermediate results. The key used here is s1+s2, but other keys are also possible (e.g. using indices)

The recursive version has exponential complexity. To further improve the performance, we can use bottom-up DP, which is O(N^4) complexity. Here we build a table isS[len][i][j], which indicates whether s1[i..i+len-1] is a scramble of s2[j..j+len-1].

Furthermore, in many cases, we found we can terminate our recursion early by pruning: i.e. by first checking if s1 and s2 have the same character set before we do recursion: if not, just terminate without recursion. This observation leads us to the following Recursion+cache+pruning version. Here the key of the cache changes to idx1sSize +idx2 + lensSize*sSize;

Code

C++ Code for Scramble String

class Solution {
private:
    bool DP_helper(string &s1, string &s2, int idx1, int idx2, int len, char isS[])
    {
        int sSize = s1.size(),i, j, k, hist[26] , zero_count =0;
        if(isS[(len*sSize+idx1)*sSize+idx2]) return isS[(len*sSize+idx1)*sSize+idx2]==1;
        bool res = false;
        
        fill_n(hist, 26, 0);
        for(k=0; k<len;++k)
        { // check if s1[idx1:idx1+len-1] and s2[idx2:idx2+len-1] have same characters
            zero_count +=  (0==hist[s1[idx1+k]-'a']) - (0== ++hist[s1[idx1+k]-'a']);
            zero_count +=  (0==hist[s2[idx2+k]-'a']) - (0== --hist[s2[idx2+k]-'a']);
        }
        if(zero_count) {isS[(len*sSize+idx1)*sSize+idx2] = 2; return false;} //if not, return directly
        if(len==1)     {isS[(len*sSize+idx1)*sSize+idx2] = 1; return true;}
        for(k=1;k<len && !res;++k) //otherwise, recursion with cache
        {
            res = res || (DP_helper(s1, s2, idx1, idx2, k, isS) && DP_helper(s1, s2, idx1+k, idx2+k, len-k, isS) );
            res = res || (DP_helper(s1, s2, idx1+len-k, idx2, k, isS) && DP_helper(s1, s2, idx1, idx2+k, len-k, isS) );
        }
        isS[(len*sSize+idx1)*sSize+idx2] = res?1:2;
        return res;
    }
public:
    bool isScramble(string s1, string s2) {
        const int sSize = s1.size();
        if(0==sSize) return true;
        char isS[(sSize+1)*sSize*sSize];
        fill_n(isS, (sSize+1)*sSize*sSize, 0);
        return DP_helper(s1, s2, 0, 0, sSize, isS);
    }
};

Java Code for Scramble String

public class Solution {
    public boolean isScramble(String s1, String s2) {
        if (s1.equals(s2)) return true; 
        
        int[] letters = new int[26];
        for (int i=0; i<s1.length(); i++) {
            letters[s1.charAt(i)-'a']++;
            letters[s2.charAt(i)-'a']--;
        }
        for (int i=0; i<26; i++) if (letters[i]!=0) return false;
    
        for (int i=1; i<s1.length(); i++) {
            if (isScramble(s1.substring(0,i), s2.substring(0,i)) 
             && isScramble(s1.substring(i), s2.substring(i))) return true;
            if (isScramble(s1.substring(0,i), s2.substring(s2.length()-i)) 
             && isScramble(s1.substring(i), s2.substring(0,s2.length()-i))) return true;
        }
        return false;
    }
}

Complexity Analysis for Scramble String LeetCode Solution

Time Complexity

O(mn)

Space Complexity

O(mn)

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