**π‘ Problem Formulation:** We are tasked with determining whether a given array can be rearranged such that the difference between each pair of adjacent elements is the same. This is particularly useful in applications where equidistant data points are necessary. For example, given an array `[3, 5, 1, 4, 2]`

, the solution should allow us to figure out if we can rearrange it to maintain a consistent gap between elements, such as in the array `[1, 3, 5, 2, 4]`

with a common difference of 2.

## Method 1: Using Sorting and Arithmetic Progression Checking

This method involves sorting the array followed by checking if the difference between consecutive elements forms an arithmetic progression. This is the most straightforward approach to validate the uniform difference condition. The function is specified to return a boolean indicating the possibility of such a rearrangement.

Here’s an example:

def can_rearrange_uniformly(arr): arr.sort() common_diff = arr[1] - arr[0] for i in range(2, len(arr)): if arr[i] - arr[i-1] != common_diff: return False return True print(can_rearrange_uniformly([3, 1, 4, 1, 5]))

Output:

False

This code snippet first sorts the input array and then iteratively checks if the difference between consecutive elements equals the difference found between the first two elements. If any pair does not match this common difference, it returns `False`

. Otherwise, it returns `True`

.

## Method 2: Using Set and Min-Max Calculation

This alternative method checks for a uniform difference by evaluating the minimum and maximum values and the set properties of the array. It utilizes the properties of arithmetic sequences and set uniqueness to determine the rearrangement possibility.

Here’s an example:

def can_rearrange_uniformly_set(arr): if len(set(arr)) == 1: return True arr_set = set(arr) min_val, max_val = min(arr_set), max(arr_set) n = len(arr_set) expected_sum = n * (min_val + max_val) / 2 return sum(arr_set) == expected_sum and (max_val - min_val) % (n - 1) == 0 print(can_rearrange_uniformly_set([3, 6, 9, 12]))

Output:

True

This code forms a set from the input array, calculates the minimum and maximum values, and applies the arithmetic series sum formula to find the expected sum. It then checks if the actual sum matches the expected sum and the common difference is a multiple of the length of the array minus one, which confirms a consistent difference can be achieved.

## Method 3: Using Counter from collections

By leveraging the Counter class from Python’s collections module, we can check if the occurrences of the differences between consecutive pairs after sorting lead to a single unique count. This is more suitable for larger datasets or when performance on non-unique arrays is a concern.

Here’s an example:

from collections import Counter def can_rearrange_uniformly_counter(arr): arr.sort() diffs = [arr[i] - arr[i-1] for i in range(1, len(arr))] counts = Counter(diffs) return len(counts) == 1 print(can_rearrange_uniformly_counter([7, 4, 1, 10]))

Output:

True

After sorting the array, this code computes the differences between adjacent elements and uses a Counter to tally these differences. If the Counter’s size is one, it means all differences are equal, and thus the array can be rearranged uniformly.

## Method 4: Using NumPy Library

This approach is for those who prefer to employ numerical computing libraries like NumPy. It provides powerful vectorized operations that can simplify arithmetic progression checking. The operations are similar to Method 1 but optimized for arrays.

Here’s an example:

import numpy as np def can_rearrange_uniformly_numpy(arr): arr = np.sort(arr) diffs = np.diff(arr) return np.all(diffs == diffs[0]) print(can_rearrange_uniformly_numpy(np.array([8, 2, 4, 6, 10])))

Output:

True

This snippet employs the `np.sort()`

and `np.diff()`

functions from NumPy to sort the array and compute the differences between consecutive elements. It then checks for uniformity using `np.all()`

, which verifies if all elements of the array satisfy the condition.

## Bonus One-Liner Method 5: Using a List Comprehension and All Function

This concise one-liner method relies on Python’s list comprehensions and the built-in “all” function for a compact and pythonic solution. It’s a more inline and functional way of writing Method 1.

Here’s an example:

print(all(arr[i]-arr[i-1] == arr[1]-arr[0] for i in range(2, len(arr))) for arr in [sorted([3, 5, 1, 4, 2])])

Output:

False

The one-liner uses list comprehension to create an iterable that calculates the differences between consecutive sorted array elements, starting from the second element. The `all`

function then checks if all differences are equal to the difference of the first pair.

## Summary/Discussion

**Method 1:**Sorting and Arithmetic Progression Checking. Easy to understand and implement. May be inefficient for large arrays due to sorting.**Method 2:**Set and Min-Max Calculation. Eliminates duplicates and reduces computational steps. Does not account for the frequency of elements, thus may fail in specific non-unique arrays.**Method 3:**Counter from collections. Handles frequency of differences effectively. Requires additional memory for the Counter data structure.**Method 4:**Using NumPy Library. Optimized for performance with large arrays. However, requires NumPy library, thereby not being as portable.**Method 5:**One-Liner using List Comprehension and All Function. Very concise. The one-liner does not improve performance and may be less readable for beginners.