Longest Palindromic Substring

The "Longest Palindromic Substring" problem is a classic challenge in computer science, particularly in string processing and dynamic programming. It involves identifying the longest substring within a given string that reads the same forwards and backwards. Let's dissect this problem using fundamental principles of algorithms and string processing.

Problem Statement:

Input: A string.
Goal: Find the longest substring which is a palindrome.

Fundamental Concepts:

String: A sequence of characters.
Substring: A contiguous sequence of characters within a string.
Palindrome: A string that reads the same forwards and backwards. For example, "racecar" is a palindrome.

Solution Approaches:

Brute Force Method:
- Idea: Check every possible substring to see if it's a palindrome.
- Time Complexity: O(n³), where n is the length of the string. This is because there are O(n²) substrings, and checking each for being a palindrome takes O(n) time.
- Space Complexity: O(1), as no extra space is required apart from the input string.
- Practicality: Not efficient for large strings due to high time complexity.
Expand Around Center:
- Idea: Consider each character (and each pair of adjacent characters for even-length palindromes) as the middle of a potential palindrome and expand outwards.
- Implementation: For each character (or character pair), expand outwards while the substring is a palindrome.
- Time Complexity: O(n²), since for each of the n characters, expansion in both directions takes O(n) time in the worst case.
- Space Complexity: O(1), only constant extra space is needed.
Dynamic Programming:
- Idea: Use a table to store the results of subproblems; a substring is a palindrome if its outer characters match and its inner substring is also a palindrome.
- Implementation: Create a boolean table dp[i][j] that is true if the substring from index i to j is a palindrome.
- Time Complexity: O(n²), since we fill a table of size n².
- Space Complexity: O(n²), for the table.

Example Implementation (Expand Around Center in Python):

Here's an example using the "Expand Around Center" method:

def longest_palindromic_substring(s):
    """
    Finds the longest palindromic substring in s.

    Args:
    s (str): The input string.

    Returns:
    str: The longest palindromic substring.
    """
    def expand_around_center(left, right):
        """Expands around the center and returns the longest palindrome."""
        while left >= 0 and right < len(s) and s[left] == s[right]:
            left -= 1
            right += 1
        return s[left + 1:right]

    longest = ""
    for i in range(len(s)):
        # Odd length palindrome
        tmp = expand_around_center(i, i)
        if len(tmp) > len(longest):
            longest = tmp

        # Even length palindrome
        tmp = expand_around_center(i, i + 1)
        if len(tmp) > len(longest):
            longest = tmp

    return longest

This implementation:

Iterates through each character in the string, treating each as the center of a potential palindrome.
Expands around the center for both odd and even length palindromes.
Uses a helper function expand_around_center to find the longest palindrome centered at the given indices.
Updates the longest palindrome found so far.

This approach efficiently finds the longest palindromic substring with significantly less time complexity compared to brute force, making it suitable for a wide range of input string sizes.

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