Approach

To effectively answer the question on implementing an algorithm to count the number of palindromic substrings in a given string, follow this structured framework:

1. Understand the Problem: Clearly define what a palindromic substring is. A substring is considered palindromic if it reads the same forwards and backwards.

2. Choose an Algorithm: Decide on the most efficient algorithm to use. Common approaches include:

• Brute Force: Check all possible substrings.

• Dynamic Programming: Use a table to store palindromic states.

• Expand Around Center: Expand potential palindromes from each character and between characters.

• Implement the Algorithm: Write the code clearly, ensuring it’s easy to follow and well-commented.

• Test the Algorithm: Validate your implementation with various test cases to ensure accuracy.

Key Points

• Definition of Palindrome: Ensure clarity on what constitutes a palindromic substring.

• Efficiency Matters: Discuss why certain algorithms are more efficient than others.

• Code Clarity: Write clean, maintainable code with comments explaining each part.

• Testing: Highlight the importance of thorough testing with edge cases.

Standard Response

Here’s a sample answer that encapsulates the above framework:

To count the number of palindromic substrings in a given string, I would implement the Expand Around Center algorithm due to its efficiency and simplicity. This method allows us to explore potential palindromes by treating each character (and the gaps between characters) as potential centers for palindromes.

Step-by-Step Implementation:

• Initialize Count: Set a counter to zero to keep track of the number of palindromic substrings.

• Expand Around Center: For each character in the string, use two pointers to expand outwards, checking for palindromic substrings:

• First, consider the character itself as the center (for odd-length palindromes).

• Then, consider the space between the character and the next one as the center (for even-length palindromes).

• Count Palindromes: Each time a palindrome is found during the expansion, increment the count.

• Return the Count: After checking all possible centers, return the total count.

Here’s how the code looks:

def count_palindromic_substrings(s: str) -> int:
 def expand_around_center(left: int, right: int) -> int:
 count = 0
 while left >= 0 and right < len(s) and s[left] == s[right]:
 count += 1
 left -= 1
 right += 1
 return count
 
 total_count = 0
 for i in range(len(s)):
 # Count odd-length palindromes
 total_count += expand_around_center(i, i)
 # Count even-length palindromes
 total_count += expand_around_center(i, i + 1)
 
 return total_count

# Example Usage
string = "abba"
print(count_palindromic_substrings(string)) # Output: 6

In this example, the string "abba" has 6 palindromic substrings: "a", "b", "b", "a", "bb", and "abba".

Tips & Variations

Common Mistakes to Avoid

• Over-Complicating the Solution: Some candidates might dive into complex algorithms or data structures when a simpler one suffices.

• Neglecting Edge Cases: Ensure to consider empty strings and single-character strings.

Alternative Ways to Answer

• Dynamic Programming: This can also be an effective solution, especially for candidates comfortable with memoization. This approach involves creating a 2D array to track palindromic states.

Role-Specific Variations

• Technical Roles: Focus on the algorithm’s time complexity (O(n^2)) and space complexity (O(1) for the Expand Around Center approach).

• Managerial Roles: Discuss the importance of efficient algorithms in product development and user experience.

• Creative Roles: Emphasize the problem-solving aspect and how algorithms can inspire innovative solutions.

Follow-Up Questions

• Can you explain why you chose the Expand Around Center method over Dynamic Programming?

• How would you optimize this algorithm further?

• What is the time complexity of your solution and why is it efficient?

• How would you handle very large strings?

By following this structured approach, candidates can ensure they provide a comprehensive, clear, and effective response to the interview question regarding palindromic substrings. This preparation not only demonstrates their technical skills