Longest Palindromic Subsequence
Given a string s, return the length of the longest palindromic subsequence in s.
Example:
bbbab
4
- The input string
sis bbbab, and we need to find the longest palindromic subsequence within it. - A palindromic subsequence can be formed by selecting characters that read the same backward as forward, such as bbbb or bbab is not a palindrome but bbb is, and single character b and a are also palindromes.
- The longest palindromic subsequence in bbbab is bbbb or bbba is not valid since it is not a palindrome, but bbbb is, so we consider bbbb as the longest palindromic subsequence.
- The length of the longest palindromic subsequence bbbb is 4, which is the output of the given input string.
Constraints:
- 1 <= len(s) <= 1000
- s consists of lowercase English letters
Background Knowledge
The Longest Palindromic Subsequence problem is a classic example of a Dynamic Programming problem. To tackle this problem, it's essential to understand the concept of a palindrome, which is a sequence that reads the same backward as forward. In the context of strings, a palindrome can be a single character, a pair of characters, or a longer sequence. The key idea is to identify the longest sequence within the given string that is a palindrome.
In Dynamic Programming, we break down complex problems into smaller sub-problems, solve each sub-problem only once, and store the solutions to sub-problems to avoid redundant computation. This approach is particularly useful for problems that have overlapping sub-problems or optimal sub-structure. The Longest Palindromic Subsequence problem exhibits both of these properties, making it an ideal candidate for a dynamic programming solution.
To understand the dynamic programming approach, it's crucial to grasp the concept of a 2D table or matrix, where each cell represents the solution to a sub-problem. In this case, the table will store the lengths of the longest palindromic subsequences for different substrings of the input string s. The state of the problem can be defined by two indices, i and j, representing the start and end of the substring, respectively. The transition between states involves comparing characters at positions i and j and updating the length of the longest palindromic subsequence accordingly.
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