Minimum and Maximum sum of absolute differences of pairs

Given an array of N integers where N is even, find the minimum and maximum sum of absolute difference of N/2 pairs formed by pairing every element with one other element.

Examples:

Input: a[] = {10, -10, 20, -40} 
Output: min_sum = 40, max_sum = 80
Explanation: Pairs selected for minimum sum 
             (-10, -40) and (10, 20) 
             min_sum = |-10 - -40| + |20 - 10| = 40 
             Pairs selected for maximum sum 
             (-10, 20) and (-40, 10) 
             max_sum = |-10 - 20| + |10 - -40| = 80

Input: a[] = {20, -10, -1, 30} 
Output: min_sum = 19, max_sum = 61 
Explanation: Pairs selected for minimum sum
             (-1, -10) and (20, 30) 
             min_sum = |-1 - -10| + |20 - 30| = 19 
             Pairs selected for maximum sum
             (-1, 30) and (-10, 20) 
             max_sum = |-1 - 30| + |-10 - 20| = 61 

Approach:
The most common observation will be that for minimum sum of differences we need the closest elements together as a pair and for the maximum sum we need the farthest elements together as a pair. So, we can simply sort the given list of elements and the closest pairs will be a[i], a[i+1], their absolute difference sum will yield us the minimum sum. The farthest will be (a[0], a[n-1]) and (a[1], a[n-2]) and so on, and their absolute difference sum will yield us the maximum-sum.

C++

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// CPP program to find minimum and maximum 
// sum of absolute differences of pairs
#include <bits/stdc++.h>
using namespace std;
  
// function to calculate minimum sum
int calculate_min_sum(int a[], int n)
{
    // sorts the array c++ stl
    sort(a, a + n);
  
    // initially min=0 and max=0
    int min_sum = 0;
  
    // traverse to find the minimum sum
    for (int i = 1; i < n; i += 2) {
  
        // the adjacent elements difference
        // will always be smaller
        min_sum += abs(a[i] - a[i - 1]);
    }
    return min_sum;
}
  
// function to calculate maximum sum
int calculate_max_sum(int a[], int n)
{
    // sorts the array c++ stl
    sort(a, a + n);
  
    int max_sum = 0;
  
    // traverse to find the maximum sum
    for (int i = 0; i < n / 2; i++) {
          
        // the farthest distant elements sum 
        // will always be maximum
        max_sum += abs(a[n - 1 - i] - a[i]);
    }
    return max_sum;
}
  
// Driver program to test above function
int main()
{
    int a[] = { 10, -10, 20, -40};
  
    int n = sizeof(a) / sizeof(a[0]);
  
    cout << "The minimum sum of pairs is " 
         << calculate_min_sum(a, n) << endl;
  
    cout << "The maximum sum of pairs is " 
         << calculate_max_sum(a, n) << endl;
  
    return 0;
}

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Java

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// Java program to find minimum and maximum 
// sum of absolute differences of pairs
import java.util.Arrays;
  
class GFG
{
    // function to calculate minimum sum
    static int calculate_min_sum(int[] a, int n)
    {
        // sorts the array c++ stl
        Arrays.sort(a);
  
        // initially min=0 and max=0
        int min_sum = 0;
  
        // traverse to find the minimum sum
        for (int i = 1; i < n; i += 2) {
  
            // the adjacent elements difference
            // will always be smaller
            min_sum += Math.abs(a[i] - a[i - 1]);
        }
        return min_sum;
    }
  
    // function to calculate maximum sum
    static int calculate_max_sum(int[] a, int n)
    {
        // sorts the array c++ stl
        Arrays.sort(a);
  
        int max_sum = 0;
  
        // traverse to find the maximum sum
        for (int i = 0; i < n / 2; i++) {
          
            // the farthest distant elements sum 
            // will always be maximum
            max_sum += Math.abs(a[n - 1 - i] - a[i]);
        }
        return max_sum;
    }
  
    // Driver program to test above function    
    public static void main (String[] args) {
    int[] a = { 10, -10, 20, -40};
  
    int n = a.length;
      
    System.out.println("The minimum sum of pairs is " +
                          calculate_min_sum(a, n)); 
  
    System.out.println("The maximum sum of pairs is " +
                           calculate_max_sum(a, n)); 
      
    }
}
  
/* This code is contributed by Mr. Somesh Awasthi */

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Python 3

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# Python 3 program to find minimum and maximum 
# sum of absolute differences of pairs
  
# function to calculate minimum sum
def calculate_min_sum( a, n):
  
    # sorts the array c++ stl
    a.sort()
  
    # initially min=0 and max=0
    min_sum = 0
  
    # traverse to find the minimum sum
    for i in range(1, n, 2):
  
        # the adjacent elements difference
        # will always be smaller
        min_sum += abs(a[i] - a[i - 1])
      
    return min_sum
  
# function to calculate maximum sum
def calculate_max_sum(a, n):
  
    # sorts the array c++ stl
    a.sort()
  
    max_sum = 0
  
    # traverse to find the maximum sum
    for i in range(n // 2):
          
        # the farthest distant elements sum 
        max_sum += abs(a[n - 1 - i] - a[i])
    return max_sum
  
# Driver Code
if __name__ == "__main__":
      
    a = [ 10, -10, 20, -40]
  
    n = len(a)
  
    print("The minimum sum of pairs is"
                calculate_min_sum(a, n))
  
    print( "The maximum sum of pairs is"
                 calculate_max_sum(a, n))
  
# This code is contributed by ita_c

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C#

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// C# program to find minimum and maximum 
// sum of absolute differences of pairs
using System;
  
class GFG
{
    // function to calculate minimum sum
    static int calculate_min_sum(int []a, int n)
    {
        // sorts the array c++ stl
        Array.Sort(a);
  
        // initially min=0 and max=0
        int min_sum = 0;
  
        // traverse to find the minimum sum
        for (int i = 1; i < n; i += 2) {
  
            // the adjacent elements difference
            // will always be smaller
            min_sum += Math.Abs(a[i] - a[i - 1]);
        }
        return min_sum;
    }
  
    // Function to calculate maximum sum
    static int calculate_max_sum(int []a, int n)
    {
        // sorts the array c++ stl
        Array.Sort(a);
  
        int max_sum = 0;
  
        // Traverse to find the maximum sum
        for (int i = 0; i < n / 2; i++) {
          
            // the farthest distant elements sum 
            // will always be maximum
            max_sum += Math.Abs(a[n - 1 - i] - a[i]);
        }
        return max_sum;
    }
  
    // Driver Code
    public static void Main () 
    {
    int []a = { 10, -10, 20, -40};
  
    int n = a.Length;
      
    Console.WriteLine("The minimum sum of pairs is " +
                            calculate_min_sum(a, n)); 
  
    Console.Write("The maximum sum of pairs is " +
                         calculate_max_sum(a, n)); 
      
    }
}
  
// This code is contributed by nitin mittal.

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PHP

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<?php
// PHP program to find minimum and maximum 
// sum of absolute differences of pairs
  
// function to calculate minimum sum
function calculate_min_sum($a, $n)
{
      
    // sorts the array c++ stl
    sort($a);
  
    // initially min=0 and max=0
    $min_sum = 0;
  
    // traverse to find the minimum sum
    for ($i = 1; $i < $n; $i += 2) 
    {
  
        // the adjacent elements difference
        // will always be smaller
        $min_sum += abs($a[$i] - 
                    $a[$i - 1]);
    }
    return $min_sum;
}
  
// function to calculate maximum sum
function calculate_max_sum($a, $n)
{
      
    // sorts the array c++ stl
    sort($a);
  
    $max_sum = 0;
  
    // traverse to find the maximum sum
    for ($i = 0; $i < $n / 2; $i++)
    {
          
        // the farthest distant elements sum 
        // will always be maximum
        $max_sum += abs($a[$n - 1 - $i] - 
                                  $a[$i]);
    }
    return $max_sum;
}
  
// Driver Code
$a = array(10, -10, 20, -40);
  
$n = sizeof($a);
  
echo("The minimum sum of pairs is "
        . calculate_min_sum($a, $n) . "\n");
  
echo("The maximum sum of pairs is "
        . calculate_max_sum($a, $n));
  
// This code is contributed by Ajit.
?>

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Output:

The minimum sum of pairs is 40
The maximum sum of pairs is 80

Time complexity : O(n log n)

This article is contributed by Raja Vikramaditya. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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