Interleaving Two Strings

Approach:

In order to solve this problem, let us solve a smaller problem first. Let’s assume that the length of the string A and B is 1 and the length of the string C is 2. Now in order to check whether C is formed by interleaving A and B or not, we consider the last character of string C and if this character matches neither with the character of the string A nor with the character of string B, then we return false. But if it matches with any of the characters either from A or from B, we check the same for the other character in C. So, let's further expand this idea, for bigger problems. So, if the last character of C, matches with either the last character of A or B, we check recursively for the second last element and then the third last element and so on, until we finally reach the base case when all the strings eventually become empty.

Steps:

1. We create a recursive function let’s say isInterleaveUtil and pass A, B, C, n1, n2, n3 as arguments, where n1,n2, and n3 are the length of A, B, and C.

bool isInterleaveUtil( A, B, C, n1, n2, n3):

1. If all the strings become empty i.e n1,n2.and n3 become 0, then simply return true.
2. If the length of C is not equal to (length of A + length of B),i.e  n1+n2 != n3 then we simply return false.
3. If the last character of both A and B matches with the last character of C, then we check for both the cases i.e. retreating A by one character and B by one character i.e call isInterleaveUtil(A, B, C, n1-1, n2, n3-1) and isInterleaveUtil(A, B, C, n1, n2-1, n3-1), and return true, if either of them returns true.
4. If the last character of C matches with the last character of A, but not with the last character of B, we move one character back in A and check recursively. I.e. call  isInterleaveUtil(A, B, C, n1-1, n2, n3-1);
5. If the last character of C matches with the last character of B, but not with the last character of A, we move one character back in B and check recursively. I.e. call  isInterleaveUtil(A, B, C, n1, n2-1, n3-1).
6. If the last character of C matches neither with the last character of A nor with the last character of B, then we simply return false.

In the previous approach, the algorithm recursively calculates and compares every possible character of A and B with C, which has many overlapping subproblems. i.e. recursive function computes the same sub-problems again and again. So we can use memoization to overcome the overlapping subproblems. To reiterate, memoization is when we store the results of all previously solved subproblems and return whether C is formed by interleaving A and B or not, from the dp[i][j] if we encounter the problem that has already been solved.

Since there are three states those are changing in the recursive function, but the change in the string C is just a result of a change in string A and string B, so we use the 2-dimensional array dp[][] to store all the subproblems where dp[i][j] will store whether C is formed by interleaving A from 0 to ith character and B from 0 to jth character or not.

Steps:

1. Create a dp array of size (N+1 * M+1) where N and M are lengths of A and B respectively. Initialize dp table to -1 initially.
2. Then we create a recursive function let’s say isInterleavingUtil and pass A, B, C, n1, n2, n3 and the dp array as arguments, where n1,n2, and n3 are the length of A, B, and C.

bool isInterleaveUtil( A, B, C, n1, n2, n3, dp):

1. If all the strings become empty i.e n1,n2.and n3 become 0, then simply return true.
2. If the length of C is not equal to (length of A + length of B),i.e  n1+n2 != n3 then we simply return false.
3. If we already solve the same subproblem for the remaining of strings A, B and C i.e. dp[n1][n2] != -1, then return dp[n1][n2].
4. If the last character of both A and B matches with the last character of C, then we check for both the cases i.e. retreating A by one character and B by one character i.e call isInterleaveUtil(A, B, C, n1-1, n2, n3-1, dp) and isInterleaveUtil(A, B, C, n1, n2-1, n3-1, dp), and store the result of both functions in dp[n1][n2], and return true, if either of them returns true.
5. If the last character of C matches with the last character of A, but not with the last character of B, we move one character back in A and check recursively. I.e. call  isInterleaveUtil(A, B, C, n1-1, n2, n3-1, dp), then store the result of function in dp[n1][n2]
6. If the last character of C matches with the last character of B, but not with the last character of A, we move one character back in B and check recursively. I.e. call  isInterleaveUtil(A, B, C, n1, n2-1, n3-1, dp), then store the result of function in dp[n1][n2].
7. If the last character of C matches neither with the last character of A nor with the last character of B, then we simply return false.

Approach:

The idea is to use a bottom-up dynamic programming approach instead of a memoization approach. In this, we use the recurrence relation of memoization approach as dp[i][j] = (dp[i-1][j] && A[i-1] == C[i+j-1]) || (dp[i][j-1] && B[j-1] == C[i+j-1]); where i is in [0, N] and j is in [0,  M], where N and M is the length of string A and string B respectively. 

Steps:

1. Create a dp array of size (N+1 * M+1) where N and M are lengths of A and B respectively. Initialize dp array to false initially.
2. If the length of C is not equal to (length of A + length of B), then we simply return false.
3. Run a loop from 0 to N and inside that run another loop from 0 to M, let’s denote the loop counters as i and j respectively.
4. Mark dp[i][j] as true, if the values of i and j are both zeroes. If the value of i is zero and j is non zero, then dp[i][j] becomes true, if dp[i][j-1] is true and the (j-1)th character of B matches with (j-1)th character of C. Similarly if j is 0, then dp[i][j] becomes true, if dp[i-1][j] is true and the (i-1)th character of A matches with (i-1)th character of C.
5. Now check for (i-1)th character of A, (j-1)th character of B and (i+j-1)th character of C.
6. If (i-1)th character of A matches with (i+j-1)th character of C and (j-1)th character of B doesn’t match with (i+j-1)th character of C, then mark dp[i][j] as dp[i-1][j].
7. If (i-1)th character of A doesn’t match with (i+j-1)th character of C and (j-1)th character of B matches with (i+j-1)th character of C, then mark dp[i][j] as dp[i][j-1].
8. If both (i-1)th character of A and (j-1)th character of B match with (i+j-1)th character of C, then mark dp[i][j] as true if either of dp[i-1][j], dp[i][j-1] is true, else mark dp[i][j] as false.
9. Finally, return dp[N][M].

Approach:

The space complexity for the last two approaches is O(N*M), however, it can be improved further to O(min(M,N)), if we use 1D dp array. The approach is the same as the previous approach except for the fact that here we use 1D dp array to store the results of the prefixes of the strings processed so far. We update dp[i] by using the value of dp[i-1] and the previous value of dp[i].

Steps:

1. If N is less than M, create a dp array of size equal N + 1, else create a dp array of size M + 1, where N is the length of A and M is the length of B. We do so in order to improve the space complexity further.
2. If the length of C is not equal to (length of A + length of B), then we simply return false.
3. Run a loop from 0 to N and inside that run another loop from 0 to M, let’s denote the loop counters as i and j respectively.
4. Mark dp[j] as true, if the values of i and j are both zeroes.
5. If the value of i is zero and j is non zero and the (j-1)th character of B matches with (j-1)th character of C, then assign dp[j] as dp[j-1].
6. If the value of j is zero and i is non zero and the (i-1)th character of A matches with (i-1)th character of C, then dp[j] remains unchanged.
7. If dp[j] is true and if (i-1)th character of A matches with (i+j-1)th character of C then it remains true. Else if dp[j] is false or dp[j-1] is true and (j-1)th character of B matches with (i+j-1)th character of C, then dp[j] is true. Else if both dp[j] and dp[j-1] are false, then dp[j] is false.
8. Finally, return the dp[M].