C++ Program for Maximum difference between groups of size two

Given an array of even number of elements, form groups of 2 using these array elements such that the difference between the group with highest sum and the one with lowest sum is maximum.
Note: An element can be a part of one group only and it has to be a part of at least 1 group. 
Examples:

Input : arr[] = {1, 4, 9, 6}
Output : 10
Groups formed will be (1, 4) and (6, 9), 
the difference between highest sum group
(6, 9) i.e 15 and lowest sum group (1, 4)
i.e 5 is 10.


Input : arr[] = {6, 7, 1, 11}
Output : 11
Groups formed will be (1, 6) and (7, 11), 
the difference between highest sum group
(7, 11) i.e 18 and lowest sum group (1, 6)
i.e 7 is 11.

Simple Approach: We can solve this problem by making all possible combinations and checking each set of combination differences between the group with the highest sum and with the lowest sum. A total of n*(n-1)/2 such groups would be formed (nC2). 
Time Complexity: O(n^3), because it will take O(n^2) to generate groups and to check against each group n iterations will be needed thus overall it takes O(n^3) time.
Efficient Approach: We can use the greedy approach. Sort the whole array and our result is sum of last two elements minus sum of first two elements.

C++

// CPP program to find minimum difference
// between groups of highest and lowest
// sums.
#include <bits/stdc++.h>
#define ll long long int
using namespace std;

ll CalculateMax(ll arr[], int n)
{
    // Sorting the whole array.
    sort(arr, arr + n);

    int min_sum = arr[0] + arr[1];
    int max_sum = arr[n-1] + arr[n-2];

    return abs(max_sum - min_sum);
}

// Driver code
int main()
{
    ll arr[] = { 6, 7, 1, 11 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << CalculateMax(arr, n) << endl;
    return 0;
}

Output:

11

Time Complexity: O (n * log n)
Further Optimization : 
Instead of sorting, we can find a maximum two and minimum of two in linear time and reduce the time complexity to O(n). 

 Below is the code for the above approach.

C++

// C++ program to find minimum difference
// between groups of highest and lowest
// sums.
#include <bits/stdc++.h>
using namespace std;

int CalculateMax(int arr[], int n)
{

    int first_min = *min_element(arr, arr + n);;
    int second_min = INT_MAX;
    for(int i = 0; i < n ; i ++)
    {
        // If arr[i] is not equal to first min
        if (arr[i] != first_min)
            second_min = min(arr[i],second_min);
    }

    int first_max = *max_element(arr, arr + n);
    int second_max = INT_MIN;
    for (int i = 0; i < n ; i ++)
    {
        // If arr[i] is not equal to first max
        if (arr[i] != first_max)
            second_max = max(arr[i],second_max);
    }

    return abs(first_max+second_max-first_min-second_min);

}

// Driver code
int main()
{
    int arr[] = { 6, 7, 1, 11 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << CalculateMax(arr, n) << endl;
    return 0;
}

// This code is contributed by Aman Kumar

Output

11

Time Complexity: O(N)
Auxiliary Space: O(1)

Please refer complete article on Maximum difference between groups of size two for more details!