Python Program for Longest Common Subsequence

Last Updated : 07 Jun, 2022

LCS Problem Statement: Given two sequences, find the length of longest subsequence present in both of them. A subsequence is a sequence that appears in the same relative order, but not necessarily contiguous. For example, “abc”, “abg”, “bdf”, “aeg”, ‘”acefg”, .. etc are subsequences of “abcdefg”. So a string of length n has 2^n different possible subsequences. It is a classic computer science problem, the basis of diff (a file comparison program that outputs the differences between two files), and has applications in bioinformatics. Examples: LCS for input Sequences “ABCDGH” and “AEDFHR” is “ADH” of length 3. LCS for input Sequences “AGGTAB” and “GXTXAYB” is “GTAB” of length 4. Let the input sequences be X[0..m-1] and Y[0..n-1] of lengths m and n respectively. And let L(X[0..m-1], Y[0..n-1]) be the length of LCS of the two sequences X and Y. Following is the recursive definition of L(X[0..m-1], Y[0..n-1]). If last characters of both sequences match (or X[m-1] == Y[n-1]) then L(X[0..m-1], Y[0..n-1]) = 1 + L(X[0..m-2], Y[0..n-2]) If last characters of both sequences do not match (or X[m-1] != Y[n-1]) then L(X[0..m-1], Y[0..n-1]) = MAX ( L(X[0..m-2], Y[0..n-1]), L(X[0..m-1], Y[0..n-2])

Python3

   # A Naive recursive Python implementation of LCS problem
 
def lcs(X, Y, m, n):
 
    if m == 0 or n == 0:
       return 0;
    elif X[m-1] == Y[n-1]:
       return 1 + lcs(X, Y, m-1, n-1);
    else:
       return max(lcs(X, Y, m, n-1), lcs(X, Y, m-1, n));
 
 
# Driver program to test the above function
X = "AGGTAB"
Y = "GXTXAYB"
print ("Length of LCS is ", lcs(X, Y, len(X), len(Y)))
 
  

Output:

Length of LCS is  4

Time Complexity: O(2ⁿ)

Auxiliary Space: O(n)

Following is a tabulated implementation for the LCS problem.

Python3

   # Dynamic Programming implementation of LCS problem 
 
def lcs(X, Y): 
    # find the length of the strings 
    m = len(X) 
    n = len(Y) 
 
    # declaring the array for storing the dp values 
    L = [[None]*(n + 1) for i in range(m + 1)] 
 
    """Following steps build L[m + 1][n + 1] in bottom up fashion 
    Note: L[i][j] contains length of LCS of X[0..i-1] 
    and Y[0..j-1]"""
    for i in range(m + 1): 
        for j in range(n + 1): 
            if i == 0 or j == 0 : 
                L[i][j] = 0
            elif X[i-1] == Y[j-1]: 
                L[i][j] = L[i-1][j-1]+1
            else: 
                L[i][j] = max(L[i-1][j], L[i][j-1]) 
 
    # L[m][n] contains the length of LCS of X[0..n-1] & Y[0..m-1] 
    return L[m][n] 
# end of function lcs 
 
 
# Driver program to test the above function 
X = "AGGTAB"
Y = "GXTXAYB"
print("Length of LCS is ", lcs(X, Y)) 
 
# This code is contributed by Nikhil Kumar Singh(nickzuck_007) 
 
  

Output:

Length of LCS is  4

Time Complexity: O(n*m)

Auxiliary Space: O(n*m)

Please refer complete article on Dynamic Programming | Set 4 (Longest Common Subsequence) for more details!

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