langestpalindromicsubsequence(dp)

The longest palindromic subsequence problem can be solved using dynamic programming. Here is an explanation of each step:

Create a table or matrix, 'dp', with dimensions (n x n), where 'n' is the length of the input string.
Initialize the diagonal elements of the matrix with 1, as every single character is a palindrome of length 1.
Traverse the matrix diagonally, from the bottom left to the top right.
For each cell (i, j), starting from the bottom, calculate the length of the longest palindromic subsequence considering the substring from index 'i' to index 'j'.
If the characters at indices 'i' and 'j' are the same, the length of the longest palindromic subsequence will be 2 plus the length of the longest palindromic subsequence for the substring between indices 'i+1' and 'j-1'.
If the characters at indices 'i' and 'j' are different, the length of the longest palindromic subsequence will be the maximum of the longest palindromic subsequence for the substring between indices 'i+1' and 'j' or the substring between indices 'i' and 'j-1'.
Continue traversing the matrix diagonally and calculating the lengths of the longest palindromic subsequences for all substrings.
Finally, the length of the longest palindromic subsequence for the entire input string will be stored in the top-right cell of the matrix.
To retrieve the actual longest palindromic subsequence, start from the top-right cell and traverse the matrix diagonally, following the rules:
If the characters at indices 'i' and 'j' are the same, include the character at index 'i' in the subsequence.
If the characters at indices 'i' and 'j' are different, move to the cell with the maximum value among the adjacent cells (either to the left or to the bottom).
Repeat this process until you reach the bottom-left cell, which represents the longest palindromic subsequence for the entire input string.

That's it! This dynamic programming approach efficiently solves the longest palindromic subsequence problem in C++.