Minimum Sum Of Absolute Difference Of Pairs Of Two Arrays In C

Minimizing the sum of absolute differences between elements of two arrays is a common optimization problem with applications in various fields. The goal is to pair elements from two arrays such that the total sum of the absolute differences between each pair is as small as possible. In this article, you will learn an efficient approach to solve this problem in C, focusing on the core concept of sorting.

Problem Statement

Given two arrays, A and B, each containing N integer elements, the objective is to find a permutation of array B (or A) such that the sum of the absolute differences |A[i] - B[i]| for all i from 0 to N-1 is minimized. This problem requires an optimal pairing strategy to achieve the smallest possible total difference.

For instance, if A = {4, 1} and B = {2, 3}, we need to find pairings that minimize |A[i] - B[i]|. Possible pairings:

• (4,2), (1,3): |4-2| + |1-3| = 2 + 2 = 4
• (4,3), (1,2): |4-3| + |1-2| = 1 + 1 = 2

The minimum sum here is 2.

Example

Consider two arrays:

• Array A: {4, 1, 8, 7}
• Array B: {2, 3, 6, 5}

The minimum sum of absolute differences for these arrays, achieved by sorting them first and then summing the absolute differences of corresponding elements, would be 6.

Background & Knowledge Prerequisites

To understand and implement the solution for this problem, readers should have a basic grasp of:

• C Language Fundamentals: Variables, arrays, loops, and functions.
• Standard Library Functions: Specifically qsort from stdlib.h for sorting arrays and abs from math.h (or stdlib.h for integers).
• Pointers: A basic understanding of how pointers work in C, especially when using qsort.
• Sorting Algorithms Concept: An intuitive understanding that sorting helps organize data for efficient processing.

Use Cases or Case Studies

This problem, or variations of it, arises in several practical scenarios:

• Resource Allocation: Matching tasks with available resources where each task has a specific requirement and each resource has a specific capability, aiming to minimize the "mismatch" cost.
• Data Matching: In data analysis, pairing similar data points from two different datasets to minimize discrepancies, for example, matching two sensor readings taken at slightly different times.
• Production Planning: Assigning workers to machines, where each worker has a skill level and each machine has a complexity level, to minimize the total effort difference.
• Supply Chain Optimization: Pairing demand points with supply sources to minimize transportation costs based on location differences.
• Image Processing: In image comparison or restoration, minimizing the sum of absolute differences between pixel values of two images to quantify similarity or error.

Solution Approaches

The most efficient and standard approach to minimize the sum of absolute differences between paired elements of two arrays is to sort both arrays.

Sorting Both Arrays

The mathematical intuition behind this approach is that to minimize the sum sum(|A[i] - B[i]|), you should pair elements that are "close" to each other in value. This is best achieved by sorting both arrays. If both arrays are sorted, pairing A[i] with B[i] for all i ensures that smaller elements are paired with smaller elements and larger elements with larger elements, effectively minimizing the individual absolute differences, and consequently, their sum.

// Minimum Sum of Absolute Difference of Pairs
#include <stdio.h>  // For printf
#include <stdlib.h> // For qsort, abs
#include <math.h>   // For fabs (though abs is sufficient for integers)

// Comparison function for qsort
int compareIntegers(const void *a, const void *b) {
    return (*(int*)a - *(int*)b);
}

int main() {
    // Step 1: Define two arrays of equal size
    int arrayA[] = {4, 1, 8, 7};
    int arrayB[] = {2, 3, 6, 5};
    int n = sizeof(arrayA) / sizeof(arrayA[0]); // Calculate the size of the arrays

    printf("Original Array A: ");
    for (int i = 0; i < n; i++) {
        printf("%d ", arrayA[i]);
    }
    printf("\\n");

    printf("Original Array B: ");
    for (int i = 0; i < n; i++) {
        printf("%d ", arrayB[i]);
    }
    printf("\\n\\n");

    // Step 2: Sort both arrays using qsort
    // qsort(base, num_elements, size_of_each_element, comparison_function)
    qsort(arrayA, n, sizeof(int), compareIntegers);
    qsort(arrayB, n, sizeof(int), compareIntegers);

    printf("Sorted Array A: ");
    for (int i = 0; i < n; i++) {
        printf("%d ", arrayA[i]);
    }
    printf("\\n");

    printf("Sorted Array B: ");
    for (int i = 0; i < n; i++) {
        printf("%d ", arrayB[i]);
    }
    printf("\\n\\n");

    // Step 3: Calculate the minimum sum of absolute differences
    long long min_abs_diff_sum = 0; // Use long long for sum to prevent overflow

    for (int i = 0; i < n; i++) {
        min_abs_diff_sum += abs(arrayA[i] - arrayB[i]);
    }

    // Step 4: Print the result
    printf("Minimum sum of absolute differences: %lld\\n", min_abs_diff_sum);

    return 0;
}

Run

Sample Output

Original Array A: 4 1 8 7 
Original Array B: 2 3 6 5 

Sorted Array A: 1 4 7 8 
Sorted Array B: 2 3 5 6 

Minimum sum of absolute differences: 6

Stepwise Explanation

1. Include Headers: The program starts by including necessary headers: stdio.h for input/output, stdlib.h for qsort and abs, and math.h (though abs from stdlib.h is sufficient for integers, fabs is for doubles).
2. Comparison Function: A custom comparison function compareIntegers is defined. This is required by qsort to know how to order elements. It takes two const void pointers, casts them to int, dereferences them, and returns their difference.
3. Array Initialization: Two integer arrays, arrayA and arrayB, are initialized with example values. The size n is calculated dynamically.
4. Display Original Arrays: The program prints the content of the original arrays for reference.
5. Sorting Arrays: Both arrayA and arrayB are sorted in ascending order using qsort. qsort is a generic sorting function that requires the base address of the array, the number of elements, the size of each element, and a pointer to a comparison function.
6. Display Sorted Arrays: The program then prints the content of the sorted arrays to show the effect of qsort.
7. Calculate Minimum Sum: A long long variable minabsdiffsum is initialized to 0. A loop iterates from i = 0 to n-1, calculating the absolute difference between arrayA[i] and arrayB[i] using abs() and adding it to minabsdiffsum. Using long long for the sum is a good practice to prevent potential integer overflow if the differences and array sizes are large.
8. Print Result: Finally, the calculated minabsdiff_sum is printed to the console.

Conclusion

The problem of finding the minimum sum of absolute differences of pairs between two arrays is efficiently solved by simply sorting both arrays. This intuitive approach leverages the property that pairing elements of similar magnitude minimizes their absolute difference, and when applied across all pairs after sorting, results in the globally minimal sum. This method is straightforward to implement and computationally efficient, making it a preferred solution for this type of optimization.

Summary

• To minimize the sum of absolute differences between paired elements of two arrays, sort both arrays independently.
• After sorting, iterate through the arrays and sum the absolute differences of elements at corresponding indices.
• This approach ensures that elements of similar values are paired, leading to the smallest possible total sum of differences.
• The C standard library function qsort() is ideal for sorting the arrays efficiently.