Minimum and Maximum sum of absolute differences of pairs

Last Updated : 20 Jul, 2022

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.

Implementation:

C++

   // 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;
}
 
  

Java

   // 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 */
 
  

Python3

   # 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
 
  

C#

   // 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.
 
  

PHP

   <?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.
?>
 
  

Javascript

   <script>
 
// Javascript 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
    a.sort();
 
    // Initially min=0 and max=0
    let min_sum = 0;
 
    // Traverse to find the minimum sum
    for(let 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
function calculate_max_sum(a, n)
{
     
    // Sorts the array c++ stl
    a.sort();
 
    let max_sum = 0;
 
    // Traverse to find the maximum sum
    for(let i = 0; i < parseInt(n / 2, 10); 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
let a = [ 10, -10, 20, -40 ];
let n = a.length;
   
document.write("The minimum sum of pairs is " +
               calculate_min_sum(a, n) + "</br>"); 
 
document.write("The maximum sum of pairs is " + 
               calculate_max_sum(a, n)); 
                
// This code is contributed by decode2207
 
</script>
 
  

Output

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

Time complexity : O(n log n)

Maximum sum of absolute difference of any permutation

GeeksforGeeks

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Article Tags :

Arrays
DSA

Practice Tags :

Arrays

Minimum and Maximum sum of absolute differences of pairs

C++

Java

Python3

C#

PHP

Javascript

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