Maximum Scalar Product Of Two Vectors C Program

In this article, you will learn how to calculate the maximum scalar product of two vectors in C programming. We will explore the underlying mathematical principle and implement an efficient solution.

Problem Statement

The scalar product (or dot product) of two vectors, say A and B, of the same dimension N, is calculated as the sum of the products of their corresponding components: A[0]B[0] + A[1]B[1] + ... + A[N-1]B[N-1]. The problem is to find the maximum possible scalar product when we can reorder the elements within each vector independently before performing the dot product.

Consider vectors A = {1, 3} and B = {2, 4}. If we sort both vectors in ascending order: A becomes {1, 3} B becomes {2, 4} The scalar product would be (1 2) + (3 * 4) = 2 + 12 = 14.

If we sort A ascending and B descending: A becomes {1, 3} B becomes {4, 2} The scalar product would be (1 * 4) + (3 * 2) = 4 + 6 = 10.

The maximum scalar product for A = {1, 3} and B = {2, 4} is 14.

Background & Knowledge Prerequisites

To understand and implement the solution, you should have a basic understanding of:

• C Programming Basics: Variables, data types, loops (for), arrays.
• Functions: How to define and call functions.
• Pointers: A rudimentary understanding of how qsort uses pointers.
• Sorting Algorithms: The concept of sorting arrays. We will use the standard library qsort function.

Use Cases or Case Studies

Finding the maximum scalar product has applications in various fields:

• Optimization Problems: In resource allocation, matching tasks (vector A) with resources (vector B) to maximize overall efficiency or profit.
• Machine Learning: For example, in feature selection or weighting, where some features contribute more to a model's output and are paired with higher weights.
• Financial Modeling: Pairing investments with different risk/return profiles to maximize portfolio returns under certain constraints.
• Signal Processing: Optimizing signal correlation by aligning sequences.
• Economics: Matching supply and demand where different goods have varying values and costs, aiming to maximize utility.

Solution Approaches

The key insight for maximizing the scalar product is derived from a mathematical rearrangement inequality. To get the maximum scalar product, you must pair the largest element of one vector with the largest element of the other, the second largest with the second largest, and so on. This is achieved by sorting both vectors in the same order (either both ascending or both descending) and then multiplying corresponding elements.

Approach 1: Sorting Both Vectors in Ascending Order

This approach leverages the rearrangement inequality principle. By sorting both vectors in non-decreasing order, we ensure that larger values in one vector are multiplied by larger values in the other, leading to the maximum possible sum of products.

One-line summary: Sort both input vectors in ascending order, then compute the dot product of the sorted vectors.

// Maximum Scalar Product
#include <stdio.h>
#include <stdlib.h> // Required for qsort

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

// Function to calculate maximum scalar product
long long maxScalarProduct(int arr1[], int arr2[], int n) {
    // Step 1: Sort both arrays in ascending order
    qsort(arr1, n, sizeof(int), compareIntegers);
    qsort(arr2, n, sizeof(int), compareIntegers);

    // Step 2: Calculate the scalar product
    long long product = 0;
    for (int i = 0; i < n; i++) {
        product += (long long)arr1[i] * arr2[i];
    }
    return product;
}

int main() {
    // Step 1: Define input vectors and their size
    int arr1[] = {1, 2, 3};
    int arr2[] = {4, 5, 6};
    int n1 = sizeof(arr1) / sizeof(arr1[0]);
    int n2 = sizeof(arr2) / sizeof(arr2[0]);

    if (n1 != n2) {
        printf("Error: Vectors must have the same dimension.\\n");
        return 1;
    }

    // Step 2: Calculate and print the maximum scalar product
    long long result = maxScalarProduct(arr1, arr2, n1);
    printf("The maximum scalar product is: %lld\\n", result);
    
    // Another example
    int arr3[] = { -1, -2, -3 };
    int arr4[] = { 1, 2, 3 };
    int n3 = sizeof(arr3) / sizeof(arr3[0]);
    
    result = maxScalarProduct(arr3, arr4, n3);
    printf("The maximum scalar product for {-1, -2, -3} and {1, 2, 3} is: %lld\\n", result);

    return 0;
}

Run

Sample Output:

The maximum scalar product is: 32
The maximum scalar product for {-1, -2, -3} and {1, 2, 3} is: -10

Stepwise Explanation:

1. Include Headers: We include stdio.h for input/output and stdlib.h for the qsort function.
2. Comparison Function compareIntegers: This helper function is required by qsort. It takes two const void pointers, casts them to int, dereferences them, and returns their difference. A positive result means a is greater than b, a negative means a is smaller, and zero means they are equal. This behavior sorts in ascending order.
3. maxScalarProduct Function:
   • It takes two integer arrays (arr1, arr2) and their size n.

1. main Function:
   • Two example arrays, arr1 and arr2, are defined.

• The scalar product is the sum of products of corresponding elements in two vectors.
• To maximize the scalar product, reorder elements such that larger values in one vector are paired with larger values in the other.
• The most efficient way to achieve this pairing is to sort both vectors in the same* order (e.g., both ascending).
• After sorting, iterate through the vectors, multiplying corresponding elements and summing the results.
• Use long long for the product sum to avoid integer overflow, especially with larger numbers or vector sizes.