# 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.

Implementation:

## C++

 `// CPP program to find minimum and maximum ` `// sum of absolute differences of pairs` `#include ` `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

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## Javascript

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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)

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