Longest Palindromic Substring

This challenge asks you to find the longest palindromic substring within a given string. Palindromes are sequences that read the same forwards and backward (e.g., "madam", "racecar"). Identifying the longest palindromic substring is a common problem in string manipulation and has applications in areas like bioinformatics and text analysis.

Problem Description

You are given a string s consisting of lowercase English letters. Your task is to find the longest substring of s that is a palindrome. A substring is a contiguous sequence of characters within a string.

What needs to be achieved:

• Identify all possible substrings of the input string s.
• Determine which of these substrings are palindromes.
• Find the longest palindrome among all identified palindromic substrings.
• Return the longest palindromic substring.

Key Requirements:

• The solution should handle empty strings and strings with a single character correctly.
• The solution should be efficient in terms of time complexity.
• The returned substring must be contiguous.

Expected Behavior:

The function should take a string s as input and return a string representing the longest palindromic substring found within s. If multiple palindromic substrings have the same maximum length, return any one of them.

Edge Cases to Consider:

• Empty input string: Return an empty string.
• Single-character input string: Return the input string itself.
• String with no palindromic substrings longer than 1 character: Return the first character of the string.
• Strings with repeating characters (e.g., "aaaaa").
• Strings with mixed case (although the problem specifies lowercase, consider how your solution would handle uppercase).

Examples

Example 1:

Input: "babad"
Output: "bab"
Explanation: "bab" and "aba" are both palindromes of length 3. The function can return either one.

Example 2:

Input: "cbbd"
Output: "bb"
Explanation: "bb" is the longest palindromic substring.

Example 3:

Input: ""
Output: ""
Explanation: Empty string input results in an empty string output.

Example 4:

Input: "a"
Output: "a"
Explanation: Single character input results in the same character being returned.

Example 5:

Input: "bananas"
Output: "anana"
Explanation: "anana" is the longest palindromic substring.

Constraints

• The length of the input string s will be between 0 and 1000 characters, inclusive.
• The input string s will only contain lowercase English letters.
• The time complexity of your solution should ideally be O(n^2), where n is the length of the string. While O(n) solutions exist, an O(n^2) solution is acceptable for this challenge.
• The space complexity of your solution should be O(1) or O(n).

Notes

Consider using dynamic programming or the "expand around center" approach to solve this problem efficiently. The "expand around center" approach involves iterating through each character in the string and treating it as the center of a potential palindrome, expanding outwards to check for palindromes of odd and even lengths. Dynamic programming involves creating a table to store whether a substring is a palindrome or not, building up the table from smaller substrings to larger ones. Think about how to optimize your solution to avoid redundant calculations.