Manacher's Algorithm

Introduction:

The problem of finding the longest palindromic substring is a classic problem in the field of string algorithms. A palindrome is a word, phrase, number, or other sequence of characters which reads the same forward and backward (ignoring spaces, punctuation, and capitalization). For example, “racecar” and “level” are palindromes.

The task of finding the longest palindromic substring can be applied in various fields such as bioinformatics where it is used to identify palindromic repeats in DNA sequences, and natural language processing where it is used in spell-checking and natural language understanding.

The traditional brute force approach to solving this problem has a time complexity of O(n^3) and dynamic programming approach has a time complexity of O(n^2), where n is the length of the input string. While these approaches work for smaller inputs, they become increasingly slow and inefficient for larger inputs. This is where Manacher’s algorithm comes in, which is a linear time algorithm for solving this problem.

Overview of Traditional Approaches:

The traditional brute force approach to solving the problem of finding the longest palindromic substring involves checking all possible substrings of the input string to see if they are palindromes and keeping track of the longest one found. This approach has a time complexity of O(n^3), where n is the length of the input string, making it inefficient for larger inputs.

Dynamic Programming is another approach to solve the problem, which has a time complexity of O(n^2). The idea is to create a table to store the result of sub-problems and use it to build the solution of the main problem. The table is filled in a bottom-up manner and each cell of the table represents whether a substring is palindrome or not.

Both of these approaches are quite inefficient for larger inputs and Manacher’s algorithm is a more efficient approach to solving the problem.

Introduction to Manacher’s Algorithm:

Manacher’s algorithm, also known as the “linear time longest palindromic substring algorithm,” is a linear time algorithm for finding the longest palindromic substring of a given string. The algorithm was invented by Gary Manacher in the 1970s.

The key insight of Manacher’s algorithm is that it converts even-length palindromes to odd-length palindromes by inserting a special character, often referred to as a “guard,” between each character of the input string. This allows the algorithm to easily check for palindromes with an even length, as well as palindromes with an odd length, with the same approach.

Step-by-Step Walkthrough:

1. First, we insert a special character, often referred to as a “guard,” between each character of the input string. For example, the string “abccba” would be converted to “#a#b#c#c#b#a#”.
2. Next, we create an array “P” to store the lengths of palindromes centered at each character of the modified string. We initialize all values of P to 0.
3. We then define two variables, “center” and “right,” to keep track of the center and the rightmost point of the palindrome that has been encountered so far.
4. We then iterate through each character of the modified string. For each character, we check if it is within the “right” boundary of the current palindrome. If it is, we can use the mirroring property of palindromes to determine the length of the palindrome centered at the current character. If it is not, we perform a brute force check to determine the length of the palindrome centered at the current character.
5. As we iterate through the string, we keep track of the largest palindrome encountered so far and its corresponding center and rightmost point.
6. Finally, we return the largest palindrome encountered as the longest palindromic substring of the input string.

Time Complexity Analysis:

Manacher’s algorithm has a linear time complexity of O(n), where n is the length of the input string. This is because the algorithm iterates through each character of the modified string only once, and the time spent on each iteration is constant. This makes Manacher’s algorithm much more efficient than the traditional brute force and dynamic programming approaches for larger inputs.

In summary Manacher’s algorithm is a clever and efficient way to solve the problem of finding the longest palindrome in a string. It’s based on the idea of converting even-length palindromes to odd-length palindromes and takes linear time to complete which makes it more efficient than other traditional approach.

Code Snippet in Python:

def longestPalSubstr(string):
    modifiedString = "#".join("^{}$".format(string))
    n = len(modifiedString)
    P = [0] * n
    C = 0
    R = 0
    for i in range (1,n-1):
        if i < R:
            P[i] = min(R-i, P[2*C-i])
        
        # Attempt to expand palindrome centered at i
        while modifiedString[i + 1 + P[i]] == modifiedString[i - 1 - P[i]]:
            P[i] += 1
        
        # If palindrome centered at i expand past R,
        # adjust center based on expanded palindrome.
        if i + P[i] > R:
            C, R = i, i + P[i]
    
    # Find the maximum element in P.
    maxLen, centerIndex = max((n, i) for i, n in enumerate(P))
    return string[(centerIndex  - maxLen)//2:(centerIndex  + maxLen)//2]

# Test
print(longestPalSubstr("abccba")) # Output: "abccba"

This code snippet demonstrates how Manacher’s algorithm can be implemented in Python. The function longestPalSubstr takes a string as an input and returns the longest palindrome substring.

Conclusion and Further Reading:

In this blog post, we discussed the problem of finding the longest palindromic substring and how Manacher’s algorithm can be used as an efficient solution for this problem. We went through the steps of Manacher’s algorithm and its time complexity analysis.

Manacher’s algorithm is a linear time algorithm that is more efficient than traditional approaches such as brute force and dynamic programming. It is widely used in various fields such as bioinformatics and natural language processing.

For those who are interested in learning more about Manacher’s algorithm and other related topics, there are several resources available online including the original paper by Gary Manacher, “An O(n) algorithm for the longest palindromic substring,” and other articles and tutorials on the topic.

To conclude, Manacher’s algorithm is a powerful and efficient solution to the problem of finding the longest palindrome in a string. It’s a clever way of converting even-length palindromes to odd-length palindromes and solving the problem in linear time making it suitable for large inputs and real-world applications.