Python program to find sum of absolute difference between all pairs in a list

In this article, we will learn how to find the sum of absolute differences between all pairs in a list. This involves calculating |a - b| for every pair of elements and summing the results.

Problem Statement

Given a list of numbers, we need to find the sum of absolute differences between all unique pairs. For example, with list [1, 3, 5], we calculate |1-3| + |1-5| + |3-5| = 2 + 4 + 2 = 8.

Method 1: Using Nested Loops

We can use two nested loops to iterate through all pairs and calculate their absolute differences ?

def sumPairs(numbers):
    diffs = []
    for i, x in enumerate(numbers):
        for j, y in enumerate(numbers):
            if i != j:
                diffs.append(abs(x - y))
    # Each pair is counted twice, so divide by 2
    return int(sum(diffs) / 2)

# Test the function
numbers = [22, 3, 55, 43]
result = sumPairs(numbers)
print(f"Sum of absolute differences: {result}")

Sum of absolute differences: 177

Method 2: Optimized Approach

We can avoid counting pairs twice by using proper loop indices ?

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

# Test the optimized function
numbers = [22, 3, 55, 43]
result = sumPairsOptimized(numbers)
print(f"Sum of absolute differences: {result}")

# Show step-by-step calculation
print("\nStep-by-step calculation:")
for i in range(len(numbers)):
    for j in range(i + 1, len(numbers)):
        diff = abs(numbers[i] - numbers[j])
        print(f"|{numbers[i]} - {numbers[j]}| = {diff}")

Sum of absolute differences: 177

Step-by-step calculation:
|22 - 3| = 19
|22 - 55| = 33
|22 - 43| = 21
|3 - 55| = 52
|3 - 43| = 40
|55 - 43| = 12

Method 3: Using List Comprehension

A more concise approach using list comprehension ?

def sumPairsComprehension(numbers):
    n = len(numbers)
    return sum(abs(numbers[i] - numbers[j]) 
               for i in range(n) 
               for j in range(i + 1, n))

# Test with different examples
test_cases = [
    [1, 3, 5],
    [22, 3, 55, 43],
    [10, 20, 30]
]

for numbers in test_cases:
    result = sumPairsComprehension(numbers)
    print(f"List {numbers}: Sum = {result}")

List [1, 3, 5]: Sum = 8
List [22, 3, 55, 43]: Sum = 177
List [10, 20, 30]: Sum = 40

Comparison

Conclusion

The optimized approach using proper loop indices is most efficient as it avoids counting pairs twice and uses constant space. For readability, list comprehension provides a clean one-liner solution.