Introduction

Counting palindromic substrings within a given string is a classic problem in computer science and combinatorics. A substring is palindromic if it reads the same forwards and backwards. The challenge is to develop an efficient algorithm that can count all such substrings in a string of length `n` in $O(n)$ time complexity. Traditional methods could be too slow for large strings, thus more optimized techniques are necessary.

The Manacher's Algorithm

One of the most effective solutions to this problem is an adaptation of Manacher's Algorithm. While originally designed to find the longest palindromic substring in $O(n)$ time, certain adaptations allow us to count all palindromic substrings efficiently. This approach avoids the naive method of checking all possible substrings, which would have a time complexity of $O(n^2)$.

Modified Manacher's Algorithm

The modified version of Manacher's Algorithm used for counting palindromic substrings relies on the concept of "expansion around center." Every character in the string (and every gap between characters) is treated as a potential center of a palindrome. Here's a step-by-step description:

1. Preparation Step: Transform the input string `s` by inserting a special separator character (that does not appear in `s`) between each character and also at the beginning and end. This transformation helps in handling even and odd length palindromes uniformly. For example, the string `abba` becomes `#a#b#b#a#`.
2. Initialization: Create an array `P` that will store the radius of the palindrome centered at each character in the transformed string.
3. Iterative Expansion: For each character in the transformed string, try to expand the palindrome and update the `P` array:
   • Maintain the `center` and `right` edge of the rightmost palindrome found so far.
   • Use mirror properties of palindromes and the `P` array to avoid redundant expansions.
   • Update `P[i]` appropriately, and adjust `center` and `right` if the expanded palindrome at `i` extends past `right`.
4. Count Palindromic Substrings: The number of palindromic substrings is the sum of the values in array `P`. Each `P[i]` contributes to the total count based on the transformed nature of the string.

Example Python Code

• Empty String: The function should handle an empty string input, returning 0, as there are no palindromic substrings.
• Single Character: A single character is a palindrome by itself. For a single character string, the result should be 1.
• DNA Sequence Analysis: Palindromic sequences play a role in biological processes and motif detection.
• Data Compression: Recognizing and utilizing palindromic structures can aid in data compression techniques.