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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:

  1. Input: A string.
  2. 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:

  1. 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.

  2. 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.

  3. 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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