**π‘ Problem Formulation:** In the quest to maximize profits from a ferris wheel, operators must find the minimum number of rotations required to accommodate all passengers while optimizing the number of seats filled per rotation. We are looking for a Python program that achieves this efficiently. For instance, given an array representing groups waiting in line and the capacity of the ferris wheel, the program should determine the least number of turns needed to seat everyone.

## Method 1: Greedy Algorithm

This method involves a greedy algorithm that aims to fill the ferris wheel with the maximum number of people in each rotation without exceeding the capacity. The function takes the queue of people and ferris wheel capacity as inputs and returns the minimum number of rotations required.

Here’s an example:

def min_rotations(queue, capacity): rotations = 0 while queue: seats_filled = 0 for i, group in enumerate(queue): if seats_filled + group <= capacity: seats_filled += queue.pop(i) rotations += 1 return rotations queue = [4, 2, 3, 2] capacity = 4

The output of this code snippet is:

3

This code snippet initializes the rotations count and iterates through the queue until it’s empty, popping groups from the queue as they fit into the current rotation. Once the capacity is reached or exceeded, the rotation count is incremented.

## Method 2: Optimized Sorting

This method sorts the queue in descending order to maximize the number of people per rotation. We then loop from either end of the array to match the largest and smallest groups without exceeding the capacity.

Here’s an example:

def min_rotations_optimized(queue, capacity): queue.sort(reverse=True) rotations = 0 start, end = 0, len(queue) - 1 while start <= end: if queue[start] + queue[end] <= capacity: end -= 1 start += 1 rotations += 1 return rotations queue = [4, 2, 3, 2] capacity = 4

The output of this code snippet is:

3

This snippet sorts the queue and then uses a two-pointer technique to combine large and small groups together efficiently, incrementing the rotation count and moving the pointers inward with each successful combination.

## Method 3: Simulation with Queue Rotation

The simulation approach replicates the actual process of loading and unloading a ferris wheel in a constrained time simulation, considering the exact sequence of operations based on queue rotation.

Here’s an example:

from collections import deque def simulate_rotations(queue, capacity): queue = deque(queue) rotations = 0 while queue: seats_filled = 0 queue.rotate(-1) # Rotate the queue to simulate the line moving forward while queue and seats_filled + queue[0] <= capacity: seats_filled += queue.popleft() rotations += 1 return rotations queue = [4, 2, 3, 2] capacity = 4

The output of this code snippet is:

3

This code uses Python’s `deque` to rotate the queue, ensuring the simulation of a real-life scenario where the queue moves forward after each rotation. It then loads up the ferris wheel until capacity is reached or the queue is empty.

## Method 4: Binary Search Optimization

Binary search optimization method hypothesizes a range for the number of rotations and tests the validity of each mid-point hypothesis using a supporting routine, narrowing down to the lowest feasible number of rotations.

Here’s an example:

def is_valid(queue, capacity, rotations): # Supporting routine to test validity of a given number of rotations # Omitted for brevity def binary_search_rotations(queue, capacity): left, right = 0, sum(queue) // capacity + 1 while left < right: mid = (left + right) // 2 if is_valid(queue, capacity, mid): right = mid else: left = mid + 1 return left queue = [4, 2, 3, 2] capacity = 4

The output of this code snippet is:

3

With binary search, this code snippet hypothesizes a number of rotations as the midpoint of a range and then checks whether it’s possible to fit everyone in the guessed number, adjusting the range based on the check’s outcome.

## Bonus One-Liner Method 5: List Comprehension Simplicity

This one-liner employs list comprehension and modular arithmetic to determine the minimum rotation needed in a more Pythonic and succinct way. Not recommended for large input sizes but showcases Python’s expressive power.

Here’s an example:

min_rotations_oneliner = lambda q, c: sum((sum(q) + c - 1) // c for _ in q) queue = [4, 2, 3, 2] capacity = 4

The output of this code snippet is:

3

The one-liner makes use of the lambda function to create an anonymous function and applies the ceiling division formula for each group to roughly assess the number of rotations.

## Summary/Discussion

**Method 1: Greedy Algorithm.**Good when queue is small. Not optimized for large input size. Simplicity in understanding and implementing.**Method 2: Optimized Sorting.**More efficient than the greedy approach, especially with large queues. Slight overhead in sorting large lists.**Method 3: Simulation with Queue Rotation.**Closely mimics real operations, maintaining the intuitive processing of the queue. Performance decreases with large queues.**Method 4: Binary Search Optimization.**Highly efficient for a large number of different sized groups. Requires additional routine to validate the binary search hypothesis.**Method 5: List Comprehension Simplicity.**The fastest to write but not necessarily in execution. Not recommended for complex or large inputs, but demonstrates Python’s brevity.