Algorithm Design Questions Medium
The longest palindromic subsequence problem is a classic problem in computer science that involves finding the longest subsequence of a given string that is also a palindrome. A palindrome is a string that reads the same forwards and backwards.
The problem can be solved using dynamic programming techniques. The basic idea is to build a table where each cell represents the length of the longest palindromic subsequence for a substring of the original string. The table is filled in a bottom-up manner, starting with the smallest substrings and gradually building up to the entire string.
To fill in each cell, we consider two cases: when the first and last characters of the substring are the same, and when they are different. If they are the same, we can include both characters in the palindromic subsequence, so the length of the subsequence is increased by 2. If they are different, we consider two possibilities: either we include the first character and find the longest palindromic subsequence for the remaining substring, or we include the last character and find the longest palindromic subsequence for the substring excluding the last character. We choose the maximum of these two possibilities.
Once the table is filled, the length of the longest palindromic subsequence for the entire string can be found in the top-right cell of the table. Additionally, the actual subsequence can be reconstructed by backtracking through the table.
The longest palindromic subsequence problem has various applications in algorithm design. One application is in DNA sequence analysis, where finding the longest palindromic subsequence can provide insights into the structure and function of the DNA sequence. Another application is in text processing, where finding the longest palindromic subsequence can be used for tasks such as finding the longest palindromic substring or detecting repeated patterns in a text.
Overall, the longest palindromic subsequence problem is a fundamental problem in algorithm design that can be solved efficiently using dynamic programming techniques. Its applications span across various domains, making it a valuable tool in computer science.