**π‘ Problem Formulation:** In this article, we explore the challenge of finding the longest distance between two 1’s in the binary representation of an integer. Given a non-negative integer, we convert it to binary and then assess the longest sequence of zeros flanked by ones, equating to the maximum distance. For instance, the input `9`

with binary form `1001`

has a longest distance of 3.

## Method 1: Iterative Traversal

The Iterative Traversal method converts the integer to a binary string, iterates through each character, and calculates the longest distance of zeros between two 1’s. The strength of this method is its simplicity and ease of understanding.

Here’s an example:

def max_distance_iterative(n): binary = bin(n)[2:] max_distance, current_distance = 0, 0 for bit in binary: if bit == '0': current_distance += 1 elif current_distance: max_distance = max(max_distance, current_distance) current_distance = 0 return max_distance print(max_distance_iterative(9)) # Example usage

Output: 3

This code snippet defines a function called `max_distance_iterative`

which iterates through the binary representation of the number, counting the distance of zeros between ones and returning the maximum distance encountered.

## Method 2: Regular Expression

With the Regular Expression approach, Python’s `re`

module is used to find all sequences of zeros between ones and return the length of the longest sequence. While a bit more advanced, this method is concise and leverages powerful regex capabilities.

Here’s an example:

import re def max_distance_regex(n): binary = bin(n)[2:] zero_sequences = re.findall('(?<=1)0+(?=1)', binary) return max(map(len, zero_sequences)) if zero_sequences else 0 print(max_distance_regex(9)) # Example usage

Output: 3

In this snippet, the `max_distance_regex`

function uses regex to find all sequences of zeros between ones and returns the length of the longest sequence, efficiently finding the maximum distance.

## Method 3: Binary Manipulation

Binary Manipulation involves direct manipulation of the binary bits using bit-shifting and masking. This approach might be more complex but can run faster as it operates on a bit level, avoiding conversions to strings or iterations.

Here’s an example:

def max_distance_binary(n): prev, max_dist = -1, 0 while n: next_one = (n & -n).bit_length() if prev != -1: max_dist = max(max_dist, next_one - prev - 1) prev = next_one n >>= next_one return max_dist print(max_distance_binary(9)) # Example usage

Output: 3

This code uses bitwise operations to find the distances between ones. The function `max_distance_binary`

computes the distance between consecutive one bits before moving to the next bit position, optimizing performance.

## Method 4: Using List Comprehension

The List Comprehension method constructs a list of zeros sequences lengths using Python’s list comprehension feature, offering a balance between readability and performance.

Here’s an example:

def max_distance_list_comprehension(n): return max([len(s) for s in bin(n)[2:].strip('0').split('1')]) if '1' in bin(n) else 0 print(max_distance_list_comprehension(9)) # Example usage

Output: 3

The `max_distance_list_comprehension`

function creates a list of the lengths of zero sequences between ones. One-liners can be elegant, but they may compromise readability for less experienced developers.

## Bonus One-Liner Method 5: Using a Functional Approach

This one-liner functional approach combines the functionality of `max`

and `split`

functions to achieve the result in a single line of code, highlighting Python’s capabilities for writing concise and functional code.

Here’s an example:

max_distance_functional = lambda n: max(map(len, bin(n)[2:].strip('0').split('1'))) print(max_distance_functional(9)) # Example usage

Output: 3

The single line lambda function `max_distance_functional`

computes the longest sequence of zeros between ones after converting the number to binary and stripping the leading and trailing zeros.

## Summary/Discussion

**Method 1: Iterative Traversal.**Straightforward logic. Good for beginners. Potentially less efficient for very large numbers.**Method 2: Regular Expression.**Concise and powerful. Requires familiarity with regex. Not the most performant for large inputs.**Method 3: Binary Manipulation.**Highly efficient. Operates on a bitwise level. More complex to understand and write.**Method 4: List Comprehension.**Balanced approach. Leverages Pythonβs list comprehension. Possibly less efficient for the largest numbers due to list operations.**Method 5: Functional One-Liner.**Extremely concise. Demonstrates functional programming in Python. May sacrifice readability for brevity.