**π‘ Problem Formulation:** This article explores different methods to determine if all palindromic substrings within a given string have odd lengths. The problem requires us to write a function that takes a string as input and returns a boolean value: True if every palindromic substring is of odd length, and False otherwise. For instance, given the input string “ababa”, the expected output is True since all palindromes like “aba”, “bab”, and “ababa” are of odd lengths.

## Method 1: Brute Force Checking

This method involves generating all possible substrings of the input string and checking if each substring is a palindrome. If it is, we then check if the length of that palindrome is odd. The function utilizes nested loops to consider every substring and returns False immediately if any even-length palindrome is found, reducing needless checks for optimal performance.

Here’s an example:

def all_palindromes_odd_length(s): for i in range(len(s)): for j in range(i+1, len(s)+1): substr = s[i:j] if substr == substr[::-1] and len(substr)%2 == 0: return False return True print(all_palindromes_odd_length("ababa"))

Output: True

In the given code snippet, the function `all_palindromes_odd_length()`

checks every substring of the given string. It first determines the substring, then checks if it is both a palindrome and of even length. If such a palindrome is found, it returns False; otherwise, it returns True after all checks, indicating all palindromes are of odd length.

## Method 2: Optimized Center Expansion

Center expansion is a more optimized way of finding palindromes. This method starts from each character (and pair of adjacent characters for even-length palindromes) and expands equally in both directions to check for palindromes. For each found palindrome, we verify if its length is odd. The method reduces the number of unnecessary comparisons, leading to improved performance.

Here’s an example:

def expand_around_center(s, left, right): while left >= 0 and right < len(s) and s[left] == s[right]: if (right - left + 1) % 2 == 0: return False left -= 1 right += 1 return True def all_odd_palindromes(s): for i in range(len(s)): if not expand_around_center(s, i, i) or not expand_around_center(s, i, i+1): return False return True print(all_odd_palindromes("ababa"))

Output: True

The function `all_odd_palindromes()`

iterates through the input string, using the helper function `expand_around_center()`

to check for palindromes starting at a specific character or pair of characters. If an even-length palindrome is encountered, the function returns False. Otherwise, it concludes that all palindromes are of odd length with a True result.

## Method 3: Dynamic Programming

Dynamic programming involves breaking the problem down into smaller, overlapping subproblems solved once and stored for future use. This approach constructs a table indicating whether a substring is a palindrome and then checks their lengths. It is much more efficient for larger strings where redundant checks are minimized.

Here’s an example:

def all_odd_length_palindromes(s): n = len(s) dp = [[False] * n for _ in range(n)] for i in range(n): dp[i][i] = True for start in range(n-1, -1, -1): for end in range(start+1, n): if s[start] == s[end]: if end-start == 1 or dp[start+1][end-1]: dp[start][end] = True if (end-start+1) % 2 == 0: return False return True print(all_odd_length_palindromes("ababa"))

Output: True

The function `all_odd_length_palindromes()`

initializes a 2D list to store the palindrome state of substrings and populates it while checking palindromic substrings. If an even-length palindrome is found during table construction, the function immediately returns False; otherwise, it finishes with True, indicating only odd-length palindromes were found.

## Method 4: Manacher’s Algorithm

Manacher’s Algorithm finds all palindromic substrings in linear time. This advanced technique uses the symmetry of palindromes to sequentially find longer palindromes based on previously identified shorter ones. Though complex, it’s a powerful method for handling strings with numerous palindromic substrings, especially in performance-critical applications.

Here’s an example:

# Implementation of Manacher's Algorithm is omitted due to complexity. # It would follow the algorithmic approach and include the odd-length check inside.

For Manacher’s Algorithm, specialized knowledge in string processing algorithms is required to implement it correctly. Given the problem statement, an implementation would incorporate a step to check for the lengths being odd immediately after identifying a palindrome.

## Bonus One-Liner Method 5: Using Regular Expressions

This clever one-liner uses Python’s `re`

(regular expression) module to find all palindromes and a list comprehension to filter for odd-length matches. Though not as efficient or reliable as other methods for all cases, its concise nature might be useful for quick checks or small strings.

Here’s an example:

import re def all_odd_palindromes_regex(s): return all(len(pal) % 2 for pal in re.findall(r'(?=(\w+)(?=\w\1))', s)) print(all_odd_palindromes_regex("ababa"))

Output: True

The one-liner `all_odd_palindromes_regex()`

uses a regular expression to match palindromes within the string. It then checks if each palindrome’s length is odd using a generator expression within the `all()`

function, which returns True only if all values evaluated by the expression are True.

## Summary/Discussion

**Method 1: Brute Force Checking.**Simple to understand and implement. Not scalable for long strings due to its O(n^3) time complexity, making it inefficient for large datasets.**Method 2: Optimized Center Expansion.**More efficient than brute force, with a complexity of O(n^2). It handles both odd and even-length palindromes seamlessly but may still be suboptimal for very large strings.**Method 3: Dynamic Programming.**Highly efficient for repeated queries on the same string, with O(n^2) time and space complexity. It may be overkill for single or simple queries and requires more memory for the DP table.**Method 4: Manacher’s Algorithm.**It provides the best performance with linear time complexity, O(n). However, it is complex to implement and understand, and therefore might be an over-engineered solution for simpler needs.**Method 5: Using Regular Expressions.**A concise and elegant solution, but its correctness and performance heavily depend on the regex engine and the specific string patterns, making it a less reliable choice for all use cases.