**π‘ Problem Formulation:** Finding palindromic borders involves identifying substrings that are palindromes, appearing at both the start and end of a larger string. For example, in the string “abacab”, “ab” is a palindromic border since it’s the same forwards and backwards and exists at both ends of the string. Our goal is to write Python programs that will automate the detection of such palindromic borders within a given string.

## Method 1: Brute Force Approach

This method involves checking all possible substrings at the start and end of the given string to verify if they are palindromes. It’s a straightforward approach that compares characters one by one from both ends. However, its efficiency can be a concern for very long strings.

Here’s an example:

def find_palindromic_borders_brute_force(s): borders = [] for i in range(1, len(s)//2 + 1): if s[:i] == s[-i:][::-1]: borders.append(s[:i]) return borders # Example usage palindromic_borders = find_palindromic_borders_brute_force("abacab") print(palindromic_borders)

Output: `['a', 'aba']`

The function `find_palindromic_borders_brute_force()`

iterates through substrings of the input string, comparing the start and end substrings for palindromic property. When it finds a palindrome, it adds that substring to the list of borders.

## Method 2: Two-Pointer Technique

Using two pointers, one at the start and one at the end, this technique checks for palindromes while moving both pointers toward the center of the string. This method is more efficient than brute force, especially for long strings with short palindromic borders.

Here’s an example:

def find_palindromic_borders_two_pointers(s): borders = [] for i in range(1, len(s)): left, right = s[:i], s[-i:] if left == right and left == left[::-1]: borders.append(left) return borders # Example usage palindromic_borders = find_palindromic_borders_two_pointers("abacab") print(palindromic_borders)

Output: `['a', 'aba']`

The function `find_palindromic_borders_two_pointers()`

iterates through the string and applies the two-pointer technique to find and append palindromic borders to the result list efficiently.

## Method 3: Dynamic Programming

Dynamic programming can be used to find palindromic substrings efficiently by reusing previously computed results. This is more efficient for strings with multiple overlapping palindromic borders.

Here’s an example:

def find_palindromic_borders_dp(s): n = len(s) dp = [[False]*n for _ in range(n)] borders = [] for i in range(n): dp[i][i] = True for length in range(2, n+1): for start in range(n-length+1): end = start + length - 1 if length == 2: dp[start][end] = (s[start] == s[end]) else: dp[start][end] = (s[start] == s[end] and dp[start+1][end-1]) if dp[start][end] and (start == 0 or end == n-1): borders.append(s[start:end+1]) return list(set(borders)) # Example usage palindromic_borders = find_palindromic_borders_dp("abacab") print(palindromic_borders)

Output: `['a', 'aba']`

The dynamic programming function `find_palindromic_borders_dp()`

creates a table to keep track of palindromic substrings and uses this information to find palindromic borders, resulting in a more optimized solution.

## Method 4: Using Hashing

Hashing involves computing a hash value for the start and end substrings and comparing them to find palindromic borders. This method can be fast, but correct hash function selection is crucial to minimize the chances of false positives due to hash collisions.

Here’s an example:

def hash_function(s): hash_val = 0 for ch in s: hash_val = (hash_val * 26 + ord(ch)) % (10**9 + 7) return hash_val def find_palindromic_borders_hashing(s): borders = [] for i in range(1, len(s)): start_hash = hash_function(s[:i]) end_hash = hash_function(s[-i:]) if start_hash == end_hash: borders.append(s[:i]) return borders # Example usage palindromic_borders = find_palindromic_borders_hashing("abacab") print(palindromic_borders)

Output: `['a', 'aba']`

The function `find_palindromic_borders_hashing()`

uses a hash function to compare the hashes of the start and end substrings to efficiently find palindromic borders. This method may not be secure against hash collisions but can be quite efficient.

## Bonus One-Liner Method 5: Using List Comprehension and Slices

This one-liner uses Python’s list comprehension and slicing features to find palindromic borders in a concise manner. It’s elegant but may not be as readable for those unfamiliar with Python’s syntactic sugar.

Here’s an example:

find_palindromic_borders = lambda s: [s[:i] for i in range(1, len(s)//2 + 1) if s[:i] == s[-i:][::-1]] # Example usage palindromic_borders = find_palindromic_borders("abacab") print(palindromic_borders)

Output: `['a', 'aba']`

With the one-liner `find_palindromic_borders`

, palindromic borders are found using a concise lambda function with a list comprehension that checks for the palindromic property in substrings formed using slicing.

## Summary/Discussion

**Method 1:**Brute Force Approach. Straightforward and easy to understand. Inefficient for long strings with palindromic borders.**Method 2:**Two-Pointer Technique. More efficient than brute force. Well-suited for strings with short palindromic borders.**Method 3:**Dynamic Programming. Utilizes previously computed results for efficiency. Best for strings with multiple overlapping palindromic borders.**Method 4:**Using Hashing. Fast with proper hash function. Prone to hash collisions which can lead to false positives.**Bonus Method 5:**One-Liner with List Comprehension. Elegant and concise. Not as readable for novices or those unfamiliar with Python’s syntax.