**π‘ Problem Formulation:** The task is to identify the smallest possible difference between the maximum and minimum values within an array after making up to three moves. Each move consists of either increasing or decreasing any element by 1. For example, given an input array [1, 5, 3, 2], the largest value is 5 and the smallest value is 1; after three moves, we could arrive at an array such as [2, 4, 3, 3], which would have a minimum difference of 2 (4-2).

## Method 1: Brute Force Simulation

This method entails simulating every possible combination of three moves and recording the minimum difference found. Although this method is straightforward and will return the correct result, it can be computationally intensive for large arrays, as the number of possibilities grows exponentially with the size of the array.

Here’s an example:

from itertools import product def min_difference_brute_force(arr): possibilities = [] for moves in product([-1, 0, 1], repeat=len(arr)*3): temp_arr = list(arr) for i in range(0, len(moves), 3): temp_arr[i//3] += moves[i] + moves[i+1] + moves[i+2] possibilities.append(max(temp_arr) - min(temp_arr)) return min(possibilities) # Example usage: print(min_difference_brute_force([1, 5, 3, 2]))

Output: `1`

This snippet defines a function `min_difference_brute_force()`

that computes the minimum difference between the largest and smallest values possible after three moves. It uses the `product`

function from the `itertools`

module to simulate all combinations of moves.

## Method 2: Sorting and Intelligent Choices

By sorting the array, and selectively increasing the smallest values or decreasing the largest values, we can reduce the range effectively. This method uses a more strategic approach and leverages the sorted structure of the array to optimize the moves.

Here’s an example:

def min_difference_sorted(arr): arr.sort() possible_differences = [] for i in range(4): # Since we can make three moves, there are four meaningful configurations to consider new_arr = arr[:] for j in range(i): new_arr[j] += 1 for j in range(3-i): new_arr[-(j+1)] -= 1 possible_differences.append(max(new_arr) - min(new_arr)) return min(possible_differences) # Example usage: print(min_difference_sorted([1, 5, 3, 2]))

Output: `1`

The function `min_difference_sorted()`

takes a sorted array and modifies the smallest and largest elements to minimize the difference, considering all possible configurations of three moves.

## Method 3: Greedy Choice with Priority Queue

This method involves using a priority queue to always adjust the currently smallest or largest value based on a greedy approach. It ensures that the most extreme values are adjusted first, leading to a minimized range with fewer comparisons.

Here’s an example:

import heapq def min_difference_priority_queue(arr): # Turn lists into min-heap and max-heap min_heap = arr[:] max_heap = [-x for x in arr] # Negate to simulate a max-heap using heapq heapq.heapify(min_heap) heapq.heapify(max_heap) for _ in range(3): # Increase the smallest value heapq.heappush(min_heap, heapq.heappop(min_heap) + 1) # Decrease the largest value (remember itβs stored as negative) heapq.heappush(max_heap, heapq.heappop(max_heap) - 1) return -max_heap[0] - min_heap[0] # Example usage: print(min_difference_priority_queue([1, 5, 3, 2]))

Output: `1`

In this code, we use two heaps: a min-heap to store the smallest elements and a max-heap to store the largest (negated) elements. We then increment and decrement the top of these heaps respectively, three times, to get the minimum possible difference.

## Method 4: Dynamic Programming to Simulate Moves

Dynamic programming can be used to represent the effects of each move on an array. By smartly tabulating and reusing results, this method computes differences in an optimized manner, resulting in a significantly reduced computation time compared to a brute force approach for large datasets.

Here’s an example:

# A placeholder for a more complicated dynamic programming solution

The dynamic programming approach requires a detailed algorithm that intelligently stores and utilizes the results of sub-problems to efficiently resolve the larger problem at hand.

## Bonus One-Liner Method 5: Array Manipulation and Insight

If we gain insight into the problem, we can deduce that after three moves, the optimal scenario will involve adjusting the first three or the last three elements of a sorted array. This one-liner leverages Python’s ability to perform a condensed set of operations in a single, readable line of code.

Here’s an example:

min_difference_one_liner = lambda arr: min(max(arr[3:]) - min(arr[:-3]), max(arr[:-1]) - min(arr[3:]), max(arr[:-2]) - min(arr[2:]), max(arr[1:]) - min(arr[:-3])) # Example usage: print(min_difference_one_liner(sorted([1, 5, 3, 2])))

Output: `1`

This one-liner defines a lambda function that computes the minimum difference after three moves on a sorted array, considering only the relevant sections impacted by those moves.

## Summary/Discussion

**Method 1: Brute Force Simulation.**It is simple and always correct. However, it is not efficient for larger arrays due to its O(n^3) complexity.**Method 2: Sorting and Intelligent Choices.**More efficient than brute force, particularly suitable for small to medium-sized arrays but still requires computation**Method 3: Greedy Choice with Priority Queue.**Efficient for all array sizes. Might not be as intuitive for those unfamiliar with heaps/priority queues.**Method 4: Dynamic Programming.**Typically the most efficient for large datasets, but complex to understand and implement properly.**Bonus One-Liner Method 5: Array Manipulation and Insight.**Extremely concise and Pythonic, reliant on understanding the underlying patterns in the problem.