Python Program for Longest Palindromic Subsequence | DP-12

Given a sequence, find the length of the longest palindromic subsequence in it.

As another example, if the given sequence is “BBABCBCAB”, then the output should be 7 as “BABCBAB” is the longest palindromic subsequence in it. “BBBBB” and “BBCBB” are also palindromic subsequences of the given sequence, but not the longest ones. 
1) Optimal Substructure: 
Let X[0..n-1] be the input sequence of length n and L(0, n-1) be the length of the longest palindromic subsequence of X[0..n-1]. 
If last and first characters of X are same, then L(0, n-1) = L(1, n-2) + 2. 
Else L(0, n-1) = MAX (L(1, n-1), L(0, n-2)). 
Following is a general recursive solution with all cases handled. 
Dynamic Programming Solution

Python3

# A Dynamic Programming based Python
# program for LPS problem Returns the length
# of the longest palindromic subsequence in seq
def lps(str):
    n = len(str)

    # Create a table to store results of subproblems
    L = [[0 for x in range(n)] for x in range(n)]

    # Strings of length 1 are palindrome of length 1
    for i in range(n):
        L[i][i] = 1

    # Build the table. Note that the lower
    # diagonal values of table are
    # useless and not filled in the process.
    # The values are filled in a
    # manner similar to Matrix Chain
    # Multiplication DP solution (See
    # https://www.geeksforgeeks.org/
    # dynamic-programming-set-8-matrix-chain-multiplication/
    # cl is length of substring
    for cl in range(2, n + 1):
        for i in range(n-cl + 1):
            j = i + cl-1
            if str[i] == str[j] and cl == 2:
                L[i][j] = 2
            elif str[i] == str[j]:
                L[i][j] = L[i + 1][j-1] + 2
            else:
                L[i][j] = max(L[i][j-1], L[i + 1][j]);

    return L[0][n-1]

# Driver program to test above functions
seq = "GEEKS FOR GEEKS"
n = len(seq)
print("The length of the LPS is " + str(lps(seq)))

# This code is contributed by Bhavya Jain

Output:

The length of the LPS is 7

Time Complexity :- O(n²)

Please refer complete article on Longest Palindromic Subsequence | DP-12 for more details!