Lexicographically smallest permutation having maximum sum of differences between adjacent elements

Given an array arr[] of size N, the task is to find the lexicographically smallest permutation of the given array such that the sum of difference between adjacent elements is maximum.

Examples:

Input: arr[] = {1, 2, 3, 4, 5}
Output: 5 2 3 4 1
Explanation:
Sum of difference between adjacent elements = (5 – 2) + (2 – 3) + (3 – 4) + (4 – 1) = 4.
{5, 2, 3, 4, 1} is lexicographically the smallest array and 4 is the maximum sum possible.

Input: arr[] = {3, 4, 1}
Output: 4 3 1
Explanation:
Sum of the difference between adjacent elements = (4 – 3) + (3 – 1) = 3
{4, 3, 1} is the lexicographically smallest permutation of array elements possible. Maximum sum of adjacent elements possible is 3.

Naive Approach: The simplest approach is to generate all permutations of the given array and find the sum of each permutation while keeping track of the maximum sum obtained. In the end, print the lexicographically smallest permutation with the maximum sum. 

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

Efficient Approach: The above approach can be optimized based on the following observation:

Let the required permutation of the array be {p₁, p₂, p₃, …, p_{N – 2}, p_{N – 1,} p_N}. 
Sum of the difference of the adjacent elements, S = (p₁-p₂) + (p₂-p₃) +….+ (p_{N – 2} – p_{N – 1}) + (p_{N – 1} – p_N)
                                                                             = p₁ – p_n

In order to maximize S, p₁ should be the largest element and p_N should be the smallest element in the given array arr[]. To build the lexicographically the smallest permutation, arrange the rest of the elements in increasing order. Follow the steps below to solve the problem:

• Sort the array arr[] in ascending order.
• Swap the first array element i.e., a[0] and the last array element i.e., arr[N – 1].
• After completing the above steps, print the modified array arr[].

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the lexicographically
// smallest permutation of an array such
// that the sum of the difference between
// adjacent elements is maximum
void maximumSumPermutation(vector<int>& arr)
{

    // Stores the size of the array
    int N = arr.size();

    // Sort the given array in
    // increasing order
    sort(arr.begin(), arr.end());

    // Swap the first and last array elements
    swap(arr[0], arr[N - 1]);

    // Print the required permutation
    for (int i : arr) {

        cout << i << " ";
    }
}

// Driver Code
int main()
{
    vector<int> arr = { 1, 2, 3, 4, 5 };
    maximumSumPermutation(arr);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;
class GFG
{

    // Function to find the lexicographically
    // smallest permutation of an array such
    // that the sum of the difference between
    // adjacent elements is maximum
    static void maximumSumPermutation(int[] arr)
    {

        // Stores the size of the array
        int N = arr.length;

        // Sort the given array in
        // increasing order
        Arrays.sort(arr);

        // Swap the first and last array elements
          int temp = arr[0];
          arr[0] = arr[N - 1];
        arr[N - 1] = temp;

        // Print the required permutation
        for (int i : arr) {

            System.out.print(i + " ");
        }
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 1, 2, 3, 4, 5 };
        maximumSumPermutation(arr);
    }
}

// This code is contributed by Dharanendra L V.

Python3

# Python program for the above approach

# Function to find the lexicographically
# smallest permutation of an array such
# that the sum of the difference between
# adjacent elements is maximum
def maximumSumPermutation(arr):

    # Stores the size of the array
    N = len(arr);

    # Sort the given array in
    # increasing order
    arr.sort();

    # Swap the first and last array elements
    temp = arr[0];
    arr[0] = arr[N - 1];
    arr[N - 1] = temp;

    # Print the required permutation
    for i in arr:
        print(i, end = " ");

# Driver Code
if __name__ == '__main__':
    arr = [1, 2, 3, 4, 5];
    maximumSumPermutation(arr);

# This code is contributed by 29AjayKumar

C#

// C# program to implement
// the above approach
using System;
class GFG{

  // Function to find the lexicographically
  // smallest permutation of an array such
  // that the sum of the difference between
  // adjacent elements is maximum
  static void maximumSumPermutation(int[] arr)
  {

    // Stores the size of the array
    int N = arr.Length;

    // Sort the given array in
    // increasing order
    Array.Sort(arr);

    // Swap the first and last array elements
    int temp = arr[0];
    arr[0] = arr[N - 1];
    arr[N - 1] = temp;

    // Print the required permutation
    foreach (int i in arr)
    {
      Console.Write(i + " ");
    }
  }


  // Driver Code
  public static void Main(String[] args)
  {
    int[] arr = { 1, 2, 3, 4, 5 };
    maximumSumPermutation(arr);
  }
}

// This code is contributed by sanjoy_62.

Javascript

<script>
// javascript program for the above approach

// Function to find the lexicographically
// smallest permutation of an array such
// that the sum of the difference between
// adjacent elements is maximum
function maximumSumPermutation(arr)
{

    // Stores the size of the array
    var N = arr.length;
    // Sort the given array in
    // increasing order
    arr.sort((a,b)=>a-b);
    // Swap the first and last array elements
      var temp = arr[0];
      arr[0] = arr[N - 1];
      arr[N - 1] = temp;

    // Print the required permutation
    document.write(arr);
}

// Driver Code
var arr = [ 1, 2, 3, 4, 5 ];
maximumSumPermutation(arr);

// This code is contributed by 29AjayKumar
</script>

Output:

5 2 3 4 1

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