Topic: Finding the number of index pairs such that the sum of elements from the first array is greater than the second array

Introduction:

In programming, we often come across situations where we need to compare two arrays and find certain patterns or conditions. One such common requirement is to determine the number of index pairs between two arrays where the sum of elements from the first array is greater than the sum of elements from the second array. This topic is all about solving this problem efficiently.

Problem Description:

Given two arrays array1 and array2 of equal length, we need to find the count of index pairs (i, j) such that the sum of array1[i] and array1[j] is greater than the sum of array2[i] and array2[j]. The objective is to determine the quantity of such index pairs.

Approach:

To solve this problem, we can follow a straightforward approach using nested loops, where we compare each pair of elements from both arrays and increment a counter if the sum from the first array is greater. However, this approach will have a time complexity of O(n^2), where n is the length of the arrays.

A more optimized approach is to sort the arrays in descending order and then compare the elements from the respective indices. By doing this, we can avoid unnecessary comparisons and achieve a time complexity of O(n log n), where n is the length of the arrays.

Steps:

1. Sort both array1 and array2 in descending order.
2. Initialize a counter variable to 0 to keep track of the count of index pairs where the sum from array1 is greater.
3. Iterate over the indices of the arrays using a loop.
4. Compare the sum of elements from the current indices of array1 and array2. If the sum from array1 is greater, increment the counter variable.
5. After traversing all the indices, the counter will contain the total count of index pairs where the sum from array1 is greater than array2.
6. Return the count of index pairs.

Pseudocode:

function countIndexPairs(array1, array2):
    Sort array1 in descending order
    Sort array2 in descending order
    
    Initialize counter = 0
    
    for i = 0 to length of array1 - 1:
        sum1 = array1[i] + array1[i+1]
        sum2 = array2[i] + array2[i+1]
        
        if sum1 > sum2:
            counter = counter + 1
    
    return counter

Example Usage:

array1 = [5, 3, 2, 1]
array2 = [4, 2, 1, 0]

result = countIndexPairs(array1, array2)
print("The total count of index pairs is:", result)

Output:

The total count of index pairs is: 3

Conclusion:

Determining the count of index pairs where the sum of elements from the first array is greater than the second array is a common problem in programming. By using an optimized approach and sorting the arrays, we can reduce the time complexity and solve the problem efficiently.