Find the longest subsequence of string X that is a substring of string Y

Question

I know how to use dynamic programming to solve the problem of finding either the most longest common subsequence or longest common substring given two strings. However, I am having a hard time to come up a solution for the problem of finding the longest subsequence of string X which is a substring of string Y.

Here is my brute force solution:

1. find all the subsequences of string X and sort them by length desc;
2. iterate through the sorted subsequnces, if current subsequence is a substring of Y, return the subsequence.

It works but the running time could be bad. Suppose all characters in X are unique, then there are 2^m subsequnces, where m is the length of X. I think checking if a string is a substring of Y takes O(n), where n is length of Y. So the overall running time is O(n*2^m).

Is a better way to do this, possibly via DP?

Edit:

Here is an example what I want to solve:

Y: 'BACDBDCD'
X: 'ABCD'

The answer would be 'ACD', because 'ACD' is the longest subsequence of X which is also a substring of Y.

Answer 1

Here are two ways to do it(both of them have polynomial time complexity).
1. Generate all substrings of Y(there are O(m^2) such substrings). For each substring, check if it is a subsequence of X(it can be done in linear time using greedy algorithm). This algorithm has O(n * m^2) time complexity, which is already not that bad.
2. If it is not fast enough, it is possible to achieve O(n * m) time complexity using dynamic programming. Let's define f(i, j) = the longest answer that ends in the i-th position in X and the j-th position in Y. The transitions are the following:

f(i + 1, j) = max(f(i + 1, j), f(i, j)) //skip this character in X
if X[i] == Y[j] //add this character to current answer
    f(i + 1, j + 1) = max(f(i + 1, j + 1), f(i, j) + 1)  

The initial value for f is 0 for all valid i and j. The answer is the largest value among f(n, j) for all valid j.

Answer 2

In Python you don't need Dynamic Programming for solving it. Use the flexibility of modifying for loop syntax in run time to achieve it:

current_len=0
subseq_len = 0
subseq_data=''
array1 = "ABCBDAB"
array2 = "BDCABA"
#array1="MICHAELANGELO"
#array2="HELLO"
m=len(array1)
n=len(array2)
#loop over first string array1 
#and increment index k to form new substrings of len-1
for k in range(0,m):
    start=0
    current_len = 0
    cur_seq =''
    #substring starting at k to m of array1
    for i in range(k,m):
        for j in range(start,n):
            if array1[i]==array2[j]:
                #increment length of matched subsequence
                current_len +=1
                #move forward index to point to remaining sub string array2
                start=j+1
                cur_seq = cur_seq+array1[i]
                break
        #print(k)
        #print(":"+cur_seq)
    #Check if current iteration for k produced longer match
    if subseq_len < current_len:
        subseq_len = current_len
        subseq_data = cur_seq
    enter code here

print(subseq_data)
print(subseq_len)

Answer 3

Here is the link to its solution on GFG Find length of longest subsequence of one string which is substring of another string