**π‘ Problem Formulation:** When arranging seating for an event or managing space effectively, it’s crucial to find the maximum distance between occupied and unoccupied seats. For example, given an arrangement like 1010001001, where ‘1’ represents an occupied seat and ‘0’ represents an empty seat, the maximum distance to the nearest occupied seat for any empty seat is a valuable metric. This article provides five methods to find this maximum distance using Python.

## Method 1: Linear Scan

This method involves iterating through the seat arrangement once, keeping track of the last occupied seat encountered, and calculating distances to the next occupied seat. It is efficient for a single pass through the data.

Here’s an example:

def max_distance(seats): last_occupied = -1 max_distance = 0 for i, seat in enumerate(seats): if seat == 1: if last_occupied != -1: max_distance = max(max_distance, (i - last_occupied) // 2) else: max_distance = i last_occupied = i max_distance = max(max_distance, len(seats) - 1 - last_occupied) return max_distance seats = [1,0,0,0,1,0,1] print(max_distance(seats))

Output:

2

This snippet defines a function `max_distance()`

which takes a list of seat occupancy statuses as input. The function keeps track of the last occupied seat and calculates distances as it iterates through the list. The maximum of these distances is the result, considering edge cases at the beginning and end of the seating row.

## Method 2: Using GroupBy from itertools

The itertools module provides a groupby function which can be used to group consecutive empty seats, making it easy to find their lengths and the maximum of such.

Here’s an example:

from itertools import groupby def max_distance(seats): groups = (list(g) for k, g in groupby(seats) if not k) return max((len(group) + 1) // 2 for group in groups) seats = [1,0,0,0,1,0,1] print(max_distance(seats))

Output:

2

This code uses `groupby(seats)`

to group consecutive zeros (empty seats) and calculates the distances. By taking half the length of each group of zeros (rounded up), we model the distance to the closest ‘1’ from any ‘0’ in each group.

## Method 3: Using NumPy

NumPy provides vectorized operations that can efficiently process seat arrangements represented as arrays. This method takes advantage of the speed and convenience of NumPy’s array operations to find the maximum distance.

Here’s an example:

import numpy as np def max_distance(seats): seats = np.array(seats) ones = np.where(seats == 1)[0] max_dist = np.max(np.diff(ones) // 2) first_seat_dist = ones[0] last_seat_dist = len(seats) - ones[-1] - 1 return max(first_seat_dist, last_seat_dist, max_dist) seats = [1,0,0,0,1,0,1] print(max_distance(seats))

Output:

2

The snippet converts the list of seats into a NumPy array, locates the positions of the occupied seats, then uses `np.diff(ones)`

to find the gaps between occupied seats. The maximum half-distance within the seating row and the edge cases are calculated to find the maximum distance.

## Method 4: Dynamic Programming

Dynamic programming can be used to solve the problem by breaking it down into simpler sub-problems. Each empty seat’s distance to the nearest occupied seat is calculated in relation to others, ensuring that all cases are considered.

Here’s an example:

def max_distance(seats): n = len(seats) left_dist = [n] * n right_dist = [n] * n max_dist = 0 for i in range(n): if seats[i] == 1: left_dist[i] = 0 elif i > 0: left_dist[i] = left_dist[i - 1] + 1 for i in range(n - 1, -1, -1): if seats[i] == 1: right_dist[i] = 0 elif i < n - 1: right_dist[i] = right_dist[i + 1] + 1 max_dist = max(max_dist, min(left_dist[i], right_dist[i])) return max_dist seats = [1,0,0,0,1,0,1] print(max_distance(seats))

Output:

2

The function `max_distance()`

uses two lists, `left_dist`

and `right_dist`

, to store the distances of each position from the nearest occupied seat to its left and right, respectively. These arrays are filled using a forward and backward pass. The maximum minimum distance from either side is then the answer.

## Bonus One-Liner Method 5: Using List Comprehension and max()

This concise method employs a one-liner list comprehension with the max function to compute the maximum distance between occupied seats without explicitly iterating through the list.

Here’s an example:

max_distance = lambda seats: max((len(s) + 1) // 2 for s in ''.join(map(str, seats)).strip('0').split('1') if s) seats = [1,0,0,0,1,0,1] print(max_distance(seats))

Output:

2

This one-liner encapsulates the `max_distance`

function as a lambda expression. It converts the list of seats to a string, strips the trailing zeros, and splits it by the occupied seats ‘1’. The length of each resulting group is halved to find the distance, and the max() function finds the largest.

## Summary/Discussion

**Method 1:** Linear Scan. Simple and easy to understand. It works well for uncrowded seating arrangements but can be inefficient for large datasets.

**Method 2:** Using GroupBy from itertools. Utilizes standard library features. Although shorter, it might be less intuitive and slightly slower than a raw iteration.

**Method 3:** Using NumPy. Fast and suitable for large data due to vectorized operations. It requires NumPy installation, which may be an overhead for smaller problems.

**Method 4:** Dynamic Programming. Robust and considers all possible distances. It can be overkill for this specific problem, and its two-pass approach means it’s not the most efficient.

**Method 5:** Bonus One-Liner. Extremely concise and leverages Python’s expressive syntax. However, it can be considered less readable and tricky to debug.