Manacher’s Algorithm for Finding Palindromic Substrings

Introduction

Palindrome detection is a classic problem in computer science, where the goal is to identify the longest palindromic substring within a given string. A palindrome is a sequence of characters that reads the same forward and backward, such as "racecar" or "madam". This problem has various applications, including text processing, DNA sequence analysis, and data validation.

One of the most efficient ways to solve this problem is through Manacher’s Algorithm, which finds the longest palindromic substring in linear time, O(n), where n is the length of the string. This is a significant improvement over the naive approach, which has a time complexity of O(n^2). In this blog, we will dive deep into Manacher’s Algorithm, explore its inner workings, and provide a detailed explanation with code examples.

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1. Naive Approach to Palindromic Substring Detection

Before diving into Manacher’s Algorithm, let’s first discuss the brute-force approach to detecting palindromic substrings. In this approach, we would:

1. Iterate through all possible substrings of the given string.

2. For each substring, check if it is a palindrome by comparing it with its reverse.

The time complexity of this approach is O(n^2) because there are O(n^2) possible substrings, and checking if each substring is a palindrome takes O(n) time.

This brute-force method can be slow, especially for long strings, which is why more efficient algorithms like Manacher’s Algorithm are used in practice.

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2. Introducing Manacher’s Algorithm

Manacher’s Algorithm is a linear-time algorithm for finding the longest palindromic substring in a given string. It works by expanding around potential centers of palindromes and using previously computed information to avoid redundant checks.

The key insight behind Manacher’s Algorithm is that palindromes have a center, and for each center, there is a maximum radius (the distance from the center to the boundary of the palindrome). Manacher’s Algorithm uses this property to efficiently compute the longest palindrome by maintaining an array that stores the radius of the palindrome centered at each position in the string.

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3. How Manacher’s Algorithm Works

The algorithm works by transforming the input string to handle both odd and even length palindromes uniformly. It does this by inserting special characters (such as #) between every character of the original string and at the beginning and end. This ensures that all palindromes, whether of odd or even length, can be treated as if they are of odd length.

For example, the string "racecar" is transformed into "#r#a$c#e#c#a#r#".

Steps of Manacher’s Algorithm:

1. Transform the Input String: Insert a special character (like #) between every character and at the ends of the string to handle even-length palindromes.

2. Use a Helper Array: Create an array P[] where P[i] stores the radius of the palindrome centered at position i in the transformed string.

3. Expand Around Centers: For each position in the transformed string, try to expand around the center as long as the characters match. Use previously computed palindromes to avoid unnecessary checks and reduce the time complexity.

4. Track the Rightmost Palindrome: Maintain a variable right that tracks the rightmost boundary of any palindrome found so far, and another variable center that tracks the center of that palindrome. If a palindrome extends beyond the current right boundary, update the center and right variables.

5. Extract the Longest Palindrome: Once the array P[] is populated, the largest value in P[] corresponds to the longest palindromic substring in the original string.

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4. Example Walkthrough

Let’s walk through an example to understand how Manacher’s Algorithm works.

Consider the string "babad". First, we transform it into "#b#a#b#a#d#".

From the array P[], we can extract the longest palindromic substring, which is "aba".

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5. Python Code Implementation of Manacher’s Algorithm

Here’s the Python code for Manacher’s Algorithm:

pythonCopy codedef manacher(s):
    # Step 1: Transform the string by inserting '#' between characters
    t = '#' + '#'.join(s) + '#'
    n = len(t)
    P = [0] * n  # Array to store the radius of palindromes
    center = 0  # Center of the current rightmost palindrome
    right = 0   # Right boundary of the current rightmost palindrome

    for i in range(1, n - 1):
        # Step 2: Use the mirror property to initialize P[i]
        mirror = 2 * center - i
        if i < right:
            P[i] = min(right - i, P[mirror])  # Avoid expanding beyond the right boundary

        # Step 3: Expand around the center
        while i + P[i] + 1 < n and i - P[i] - 1 >= 0 and t[i + P[i] + 1] == t[i - P[i] - 1]:
            P[i] += 1

        # Step 4: Update the center and right boundary
        if i + P[i] > right:
            center = i
            right = i + P[i]

    # Step 5: Find the maximum length palindrome
    max_len = max(P)
    center_index = P.index(max_len)
    start = (center_index - max_len) // 2  # Convert the index back to the original string
    return s[start:start + max_len]

# Example usage:
s = "babad"
longest_palindrome = manacher(s)
print(f"The longest palindromic substring is: {longest_palindrome}")

Explanation of the Code:

1. Transform the Input String: The string s is transformed by inserting # between every character, and at the start and end. This helps to handle both odd and even-length palindromes uniformly.

2. Expand Around Centers: For each position i, we try to expand around the center i as long as the characters match, and use the previously computed values in P[] to avoid redundant expansions.

3. Track the Rightmost Palindrome: The center and right variables help track the rightmost palindrome’s center and boundary. If a new palindrome extends beyond the current right boundary, we update these values.

4. Extract the Longest Palindrome: The longest palindrome is determined by finding the maximum value in the P[] array, which represents the radius of the longest palindrome centered at each position in the transformed string.

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6. Time Complexity of Manacher’s Algorithm

Manacher’s Algorithm runs in O(n) time, where n is the length of the transformed string (which is at most 2n + 1, where n is the length of the original string). The algorithm only requires a single pass through the string, and each character is processed at most once. This makes it a significant improvement over the naive O(n^2) approach.

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7. Applications of Manacher’s Algorithm

Manacher’s Algorithm is widely used in applications that require efficient palindrome detection, such as:

• Text Processing: Detecting palindromes in large text files.

• Bioinformatics: Finding palindromic sequences in DNA and protein sequences.

• Pattern Recognition: Identifying palindromic patterns in data streams.

• String Matching: Enhancing algorithms that rely on pattern matching.

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8. Conclusion

Manacher’s Algorithm is a powerful and efficient solution for finding the longest palindromic substring in linear time. By transforming the input string and leveraging previously computed information, it avoids redundant checks and ensures optimal performance. This algorithm is widely applicable in text processing, bioinformatics, and other domains that require efficient palindrome detection.

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FAQs

Q1: Why do we transform the string in Manacher’s Algorithm?
The transformation (inserting # between characters) allows the algorithm to handle both odd and even-length palindromes uniformly, simplifying the implementation.

Q2: What is the time complexity of Manacher’s Algorithm?
Manacher’s Algorithm runs in O(n) time, where n is the length of the transformed string, making it much more efficient than the brute-force approach.

Q3: Can Manacher’s Algorithm be used for substring matching?
While Manacher’s Algorithm is specifically designed for finding palindromes, it can be adapted for substring matching with some modifications.

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