###### Finding the Longest Palindromic Substring in Python

A palindromic substring is a sequence of characters that reads the same backward as forward. To determine the longest palindromic substring in a given string, one effective approach is to use the "expand around center" technique, which has a time complexity of O(n^2) and a space complexity of O(1).
Below is a Python function that implements this technique:
def longest_palindromic_substring(s: str) -> str:
if not s:
return ""
start, end = 0, 0
for i in range(len(s)):
len1 = expand_around_center(s, i, i) # Odd length palindromes
len2 = expand_around_center(s, i, i + 1) # Even length palindromes
max_len = max(len1, len2)
if max_len > end - start:
start = i - (max_len - 1) // 2
end = i + max_len // 2
return s[start:end + 1]
def expand_around_center(s: str, left: int, right: int) -> int:
while left >= 0 and right < len(s) and s[left] == s[right]:
left -= 1
right += 1
return right - left - 1

###### How It Works

1. **Function Definition**: The function *longest_palindromic_substring* takes a string *s* as input and initializes two variables, *start* and *end*, to keep track of the current longest palindromic substring's starting and ending indices.
2. **Iterating Through the String**: The function iterates over each character in the string, treating each character (and the gap between characters) as a potential center for palindromes.
3. **Expanding Around Centers**: For each character:
- The function checks for odd-length palindromes by calling *expand_around_center(s, i, i)*.
- It checks for even-length palindromes by calling *expand_around_center(s, i, i + 1)*.
4. **Updating Indices**: If a longer palindromic substring is found (i.e., *max_len*), the *start* and *end* indices are updated.
5. **Returning the Result**: The substring is extracted using the calculated *start* and *end* indices and returned as the result.
###### Example Usage

Here’s how to use the function to find the longest palindromic substring:
input_string = "babad"
result = longest_palindromic_substring(input_string)
print(f"The longest palindromic substring of '{input_string}' is '{result}'")

###### Output

For the above example, possible outputs include "bab" or "aba" since both are valid palindromic substrings of the same maximum length.
This approach provides a straightforward solution for finding the longest palindromic substring efficiently.