Efficient Algorithm for Finding the Longest Palindromic Substring: Manacher's Algorithm

When dealing with algorithms that identify the longest palindromic substring in a string, one of the most efficient methods is Manacher's Algorithm. This algorithm operates in linear time, O(n), making it significantly faster than other methods such as dynamic programming and brute force approaches. In this article, we will delve into what Manacher's Algorithm is, its key features, and provide a detailed Python implementation.

Overview of Manacher's Algorithm

Manacher's Algorithm is a technique used to find the longest palindromic substring in a string in linear time. It is particularly effective in handling both even and odd-length palindromes uniformly. This section will provide an overview of the algorithm, its key components, and the steps involved in its implementation.

Preprocessing

One of the initial steps in preprocessing is to transform the input string to handle both even and odd-length palindromes. This is typically achieved by adding special boundary characters between each character, such as the string "$T1T2T3T4$", where "$" and "$" are unique characters not present in the string.

Palindrome Length Array

Manacher's Algorithm uses an array called P where each element P[i] represents the radius of the palindrome centered at the transformed string's i-th position. This helps in efficiently expanding around centers and reducing unnecessary comparisons.

Center and Right Edge Tracking

Two variables, C and R, are crucial for tracking the current center and right edge of the known palindrome. These variables help in skipping unnecessary comparisons by utilizing previously calculated palindrome lengths. If the palindrome currently being expanded extends beyond the right edge, the center and right edge are updated accordingly.

Expand Around Centers

The core logic of Manacher's Algorithm involves expanding around each character to find the maximum length of the palindrome centered at that position. This is done by comparing the mirrored value of the current position with its possible palindrome expansion.

Implementation in Python

Here is a detailed Python implementation of Manacher's Algorithm:

def manachers_algorithm(s: str) -> str: # Preprocess the string to handle both even and odd-length palindromes T ''.join(['_' c '_' for c in s]) n len(T) P [0] * n C R 0 # Center and right edge of the current palindrome for i in range(1, n - 1): mirr 2 * C - i # Mirror of the current position # If the palindrome extends beyond the current right edge, use the mirrored value if R > i: P[i] min(R - i, P[mirr]) # Attempt to expand the palindrome centered at i while T[i 1 P[i]] T[i - 1 - P[i]]: P[i] 1 # If the palindrome expands beyond the current right edge, update the center and right edge if i P[i] > R: C, R i, i P[i] # Find the maximum element in P to determine the longest palindromic substring max_len max(P) center_index (max_len) # Adjust the start index for the original string and return the longest palindromic substring start center_index - max_len // 2 return s[start:start max_len]

Example Usage

Let's see how to use the function with an example:

s 'babad' longest_palindrome manachers_algorithm(s) print(longest_palindrome) # Output: 'bab'

Explanation of the Code

Preprocessing: The input string is transformed to handle both even and odd-length palindromes uniformly by adding special boundary characters.

P Array: The P array is filled by expanding around each character, which helps in identifying potential palindromes.

Longest Palindrome: After processing, the longest palindromic substring is determined by the maximum value in P.

Conclusion

Manacher's Algorithm is the most efficient method for finding the longest palindromic substring, achieving O(n) time complexity. This makes it significantly faster than other methods such as dynamic programming (O(n^2)) and brute force (O(n^3)).