feat: add longest palindromic substring algorithm

[ARYPROGRAMMER]

Description of Change

This PR adds a comprehensive implementation of the Longest Palindromic Substring algorithm to the dynamic programming section.

The longest palindromic substring problem finds the contiguous substring of maximum length in a given string that is also a palindrome. This is different from the existing longest palindromic subsequence implementation, where characters don't need to be contiguous.

Key Features:

• Two implementations provided:
  • Standard DP approach: O(n²) time, O(n²) space
  • Optimized expand-around-center: O(n²) time, O(1) space
• 8 comprehensive test cases covering edge cases
• Full Doxygen documentation with complexity analysis
• Real-world applications included (DNA sequencing, text processing, bioinformatics)
• Interactive examples demonstrating practical usage

Example:

• Input: "bananas" → Output: "anana" (length 5)
• Input: "racecar" → Output: "racecar" (length 7)

Checklist

• Added description of change
• Added file name matches File name guidelines
• Added tests and example, test must pass
• Added documentation so that the program is self-explanatory and educational - Doxygen guidelines
• Relevant documentation/comments is changed or added
• PR title follows semantic commit guidelines
• Search previous suggestions before making a new one, as yours may be a duplicate.
• I acknowledge that all my contributions will be made under the project's license.

Notes: Adds longest palindromic substring algorithm with DP and space-optimized implementations, 8 passing tests, and complete Doxygen documentation. Updates DIRECTORY.md accordingly.