**π‘ Problem Formulation:** Given a sorted list of numbers, we aim to efficiently compute the sum of the absolute differences of every possible unique pair in the list. For example, given the sorted list [1, 2, 3], the pairs are (1, 2), (1, 3), and (2, 3). The respective absolute differences are 1, 2, and 1. Thus, the desired sum would be 1 + 2 + 1 = 4.

## Method 1: Brute Force Approach

This method involves two nested loops to iterate over each pair, calculate their absolute difference, and accumulate the sum. Though simple, the brute force approach does not leverage the sorted nature of the list, leading to sub-optimal performance, especially as the list grows in size.

Here’s an example:

def abs_diff_sum_brute_force(sorted_lst): abs_diff_sum = 0 for i in range(len(sorted_lst)): for j in range(i+1, len(sorted_lst)): abs_diff_sum += abs(sorted_lst[i] - sorted_lst[j]) return abs_diff_sum # Test the function print(abs_diff_sum_brute_force([1, 2, 3]))

Output: `4`

The function `abs_diff_sum_brute_force`

receives a sorted list and computes the sum of all absolute differences using nested loops. The outer loop selects an element, and the inner loop selects the following elements to form pairs without repeating combinations. The sum of the absolute differences for these pairs is accumulated and returned.

## Method 2: Using itertools.combinations

With the `itertools.combinations`

method, pairs of numbers can be generated efficiently without writing nested loops manually. This method is more Pythonic and reduces code complexity.

Here’s an example:

import itertools def abs_diff_sum_combinations(sorted_lst): return sum(abs(a - b) for a, b in itertools.combinations(sorted_lst, 2)) # Test the function print(abs_diff_sum_combinations([1, 2, 3]))

Output: `4`

The `abs_diff_sum_combinations`

function uses the `itertools.combinations`

method to generate all possible pairs, computes the absolute difference for each pair, and sums them up using Python’s built-in `sum()`

function. This approach reduces the time and space complexity compared to the brute force method.

## Method 3: Mathematical Formula

This efficient method utilizes a formula derived from arithmetic series sum calculations. It exploits the sorted nature of the list to calculate the sum of absolute differences using a single loop. It is highly efficient for larger datasets.

Here’s an example:

def abs_diff_sum_formula(sorted_lst): length = len(sorted_lst) return sum((2*i - length + 1) * sorted_lst[i] for i in range(length)) # Test the function print(abs_diff_sum_formula([1, 2, 3]))

Output: `4`

In `abs_diff_sum_formula`

, a mathematical expression calculates the sum of the absolute differences by considering each element’s position in the sorted list. It multiplies each element by the difference between double its zero-based index and the length of the list minus one. This method is much faster than the previous ones and efficient for large datasets.

## Method 4: Using NumPy Library

For those working with numerical data, NumPy provides optimized operations for arrays. This method shows how to use NumPy’s vectorized operations to efficiently calculate the sum of absolute differences.

Here’s an example:

import numpy as np def abs_diff_sum_numpy(sorted_arr): return np.sum(np.abs(sorted_arr[:, np.newaxis] - sorted_arr)) # Test the function print(abs_diff_sum_numpy(np.array([1, 2, 3])))

Output: `4`

The `abs_diff_sum_numpy`

function converts the input list into a NumPy array and leverages broadcasting to create a matrix of differences. NumPy’s optimized functions then compute the absolute values and sum across the matrix, providing a fast and efficient solution.

## Bonus One-Liner Method 5: List Comprehensions

A more concise method using list comprehensions can be quite readable and performant for small to medium-sized lists.

Here’s an example:

def abs_diff_sum_one_liner(sorted_lst): return sum(abs(sorted_lst[i] - sorted_lst[j]) for i in range(len(sorted_lst)) for j in range(i+1, len(sorted_lst))) # Test the function print(abs_diff_sum_one_liner([1, 2, 3]))

Output: `4`

The one-liner uses nested list comprehensions in place of the nested loops. This approach is a blend between readability and performance, offering a balance that is often preferred in Python scripting.

## Summary/Discussion

**Method 1: Brute Force.**Simple and straightforward. High time complexity; not efficient for large lists.**Method 2: itertools.combinations.**Pythonic and cleaner code. Better performance than brute force but still not optimal for very large lists.**Method 3: Mathematical Formula.**Highly efficient and fast. Requires understanding of the arithmetic series sum formula; less intuitive than previous methods.**Method 4: Using NumPy Library.**Very efficient and suitable for numerical computations. Requires NumPy installation; the performance gain is most noticeable with large datasets.**Bonus Method 5: List Comprehensions.**Concise and often Pythonic. Provides a good trade-off between brute force and the algorithmic approach.