5 Best Ways to Find Sum of Absolute Differences in a Sorted Array in Python

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💡 Problem Formulation: We need to calculate the sum of the absolute differences between each element in a sorted array and all other elements. Given an input array such as [1, 3, 5, 7], we want to calculate an output that is the sum of absolute differences in this fashion: abs(1-3) + abs(1-5) + abs(1-7) + abs(3-5) + abs(3-7) + abs(5-7), which is 2+4+6+2+4+2=20.

Method 1: Naive Approach

The naive approach iterates over the array with two loops, calculating the absolute difference between every pair of elements and sums these differences.

Here’s an example:

def sum_of_abs_diffs(arr):
    total = 0
    for i in range(len(arr)):
        for j in range(i+1, len(arr)):
            total += abs(arr[i]-arr[j])
    return total

# Test the function
sorted_array = [1, 3, 5, 7]
print(sum_of_abs_diffs(sorted_array))

Output: 20

This code snippet defines a function sum_of_abs_diffs() that takes a sorted list and returns the sum of absolute differences between each pair of elements. It uses a nested loop to iterate through the array elements to calculate the sum total of their absolute differences.

Method 2: Using itertools.combinations

With itertools.combinations, we can easily generate all unique pairs in the array and sum their absolute differences.

Here’s an example:

import itertools

def sum_of_abs_diffs_with_combinations(arr):
    return sum(abs(a - b) for a, b in itertools.combinations(arr, 2))

# Test the function
sorted_array = [1, 3, 5, 7]
print(sum_of_abs_diffs_with_combinations(sorted_array))

Output: 20

This snippet uses itertools.combinations to generate all possible pairs (without duplication) from the array, computes their absolute differences, and then sums those values up, all in a single expression.

Method 3: Using NumPy

By leveraging the NumPy library, this method takes advantage of vectorized operations to evaluate the absolute differences efficiently.

Here’s an example:

import numpy as np

def sum_of_abs_diffs_with_numpy(arr):
    arr_np = np.array(arr)
    diffs = np.abs(arr_np - arr_np[:, None])
    return np.sum(np.triu(diffs, 1))

# Test the function
sorted_array = [1, 3, 5, 7]
print(sum_of_abs_diffs_with_numpy(sorted_array))

Output: 20

This code utilizes NumPy for high performance matrix operations. It first converts the array into a NumPy array, then calculates a matrix of absolute differences, and finally sums the upper triangle (excluding the diagonal) to obtain the desired sum.

Method 4: Mathematical Insight

Using the properties of the sorted array, we can calculate the sum in a more efficient way by understanding that each element’s contribution to the sum can be determined by its index.

Here’s an example:

def sum_of_abs_diffs_mathematical(arr):
    return sum((2 * i - len(arr) + 1) * arr[i] for i in range(len(arr)))

# Test the function
sorted_array = [1, 3, 5, 7]
print(sum_of_abs_diffs_mathematical(sorted_array))

Output: 20

In this method, the sum of absolute differences is computed by taking into account the number of elements before and after each element in the sorted array. This results in a significantly more efficient calculation in linear time.

Bonus One-Liner Method 5: Compact Code with List Comprehension

A compact one-liner Python solution using list comprehension to express the computation succinctly.

Here’s an example:

# One-liner to calculate the sum of absolute differences
sorted_array = [1, 3, 5, 7]
print(sum(abs(x - y) for i, x in enumerate(sorted_array) for y in sorted_array[i+1:]))

Output: 20

This elegant one-liner uses list comprehension to mimic the nested loop in the naive approach but in a more concise and Pythonic way, making it both easy to read and write.

Summary/Discussion

  • Method 1: Naive Approach. Simple and straightforward. Not efficient for large arrays due to O(n^2) complexity.
  • Method 2: Using itertools.combinations. Cleaner than the naive approach and uses itertools library. Performance is still O(n^2) due to combination generation.
  • Method 3: Using NumPy. Leverages fast array operations for better performance. Requires NumPy installed. More efficient for very large arrays.
  • Method 4: Mathematical Insight. Most efficient with O(n) complexity. Requires insight into the problem that might not be immediately obvious.
  • Method 5: Compact Code with List Comprehension. Pythonic and succinct. Easier to write and read but does not offer a performance improvement over the naive approach.