Arrange first N natural numbers such that absolute difference between all adjacent elements > 1

Given an integer N. The task is to find the permutation of first N natural numbers such that the absolute difference between any two consecutive numbers > 1. If no such permutation is possible then print -1.

Examples:

Input: N = 5 
Output: 5 3 1 4 2

Input: N = 3 
Output: -1 

Approach: There may be many such arrangements possible but one of the most common and greedy approach is to arrange all odd numbers in decreasing (or increasing) order and after that arrange all even numbers in decreasing (or increasing) order. Note that if N = 3 or N = 2 then there will be no such arrangement possible and if N = 1 then the sequence will consist of a single element i.e. 1.

Below is the implementation of the above approach:  

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function to print the required permutation
void arrange(int N)
{

    if (N == 1) {
        cout << "1";
        return;
    }

    // No permutation is possible
    // satisfying the given condition
    if (N == 2 || N == 3) {
        cout << "-1";
        return;
    }

    // Maximum even and odd elements < N
    int even = -1, odd = -1;
    if (N % 2 == 0) {
        even = N;
        odd = N - 1;
    }
    else {
        odd = N;
        even = N - 1;
    }

    // Print all odd elements in decreasing order
    while (odd >= 1) {
        cout << odd << " ";

        // Next element must be odd
        odd = odd - 2;
    }

    // Print all even elements in decreasing order
    while (even >= 2) {
        cout << even << " ";

        // Next element must be even
        even = even - 2;
    }
}

// Driver code
int main()
{
    int N = 5;
    arrange(N);

    return 0;
}

Java

// Java implementation of the approach
class GFG
{

// Function to print the required
// permutation
static void arrange(int N)
{
    if (N == 1)
    {
        System.out.println("1");
        return;
    }

    // No permutation is possible
    // satisfying the given condition
    if (N == 2 || N == 3)
    {
        System.out.println("-1");
        return;
    }

    // Maximum even and odd elements < N
    int even = -1, odd = -1;
    if (N % 2 == 0)
    {
        even = N;
        odd = N - 1;
    }
    else
    {
        odd = N;
        even = N - 1;
    }

    // Print all odd elements in
    // decreasing order
    while (odd >= 1)
    {
        System.out.print(odd);
        System.out.print(" ");

        // Next element must be odd
        odd = odd - 2;
    }

    // Print all even elements in
    // decreasing order
    while (even >= 2)
    {
        System.out.print(even);
        System.out.print(" ");

        // Next element must be even
        even = even - 2;
    }
}

// Driver code
public static void main(String[] args)
{
    int N = 5;
    arrange(N);
}
}

// This code is contributed
// by Akanksha Rai

Python3

# Python3 implementation of the approach

# Function to print the required permutation
def arrange(N):

    if (N == 1) :
        print("1")
        return

    # No permutation is possible
    # satisfying the given condition
    if (N == 2 or N == 3) :
        print("-1")
        return

    # Maximum even and odd elements < N
    even = -1
    odd = -1
    if (N % 2 == 0):
        even = N
        odd = N - 1
    else :
        odd = N
        even = N - 1

    # Print all odd elements in
    # decreasing order
    while (odd >= 1):
        print(odd, end = " ")

        # Next element must be odd
        odd = odd - 2

    # Print all even elements in
    # decreasing order
    while (even >= 2):
        print(even, end = " ")

        # Next element must be even
        even = even - 2

# Driver code
if __name__ == "__main__":

    N = 5
    arrange(N)

# This code is contributed by ita_c

C#

// C# implementation of the approach
using System;

class GFG
{

// Function to print the required
// permutation
static void arrange(int N)
{
    if (N == 1)
    {
        Console.WriteLine("1");
        return;
    }

    // No permutation is possible
    // satisfying the given condition
    if (N == 2 || N == 3)
    {
        Console.WriteLine("-1");
        return;
    }

    // Maximum even and odd elements < N
    int even = -1, odd = -1;
    if (N % 2 == 0)
    {
        even = N;
        odd = N - 1;
    }
    else
    {
        odd = N;
        even = N - 1;
    }

    // Print all odd elements in
    // decreasing order
    while (odd >= 1)
    {
        Console.Write(odd);
        Console.Write(" ");

        // Next element must be odd
        odd = odd - 2;
    }

    // Print all even elements in
    // decreasing order
    while (even >= 2)
    {
        Console.Write(even);
        Console.Write(" ");

        // Next element must be even
        even = even - 2;
    }
}

// Driver code
public static void Main()
{
    int N = 5;
    arrange(N);
}
}

// This code is contributed
// by Shivi_Aggarwal

PHP

<?php
// PHP implementation of the approach

// Function to print the required
// permutation
function arrange($N)
{
    if ($N == 1)
    {
        echo "1";
        return;
    }

    // No permutation is possible
    // satisfying the given condition
    if ($N == 2 || $N == 3)
    {
        echo "-1";
        return;
    }

    // Maximum even and odd elements < N
    $even = -1 ;
    $odd = -1;

    if ($N % 2 == 0)
    {
        $even = $N;
        $odd = $N - 1;
    }
    else
    {
        $odd = $N;
        $even = $N - 1;
    }

    // Print all odd elements in
    // decreasing order
    while ($odd >= 1)
    {
        echo $odd, " ";

        // Next element must be odd
        $odd = $odd - 2;
    }

    // Print all even elements in
    // decreasing order
    while ($even >= 2)
    {
        echo $even, " ";

        // Next element must be even
        $even = $even - 2;
    }
}

// Driver code
$N = 5;
arrange($N);

// This code is contributed by Ryuga
?>

Javascript

<script>

// Javascript implementation of the approach

// Function to print var the required
// permutation
function arrange(N)
{
    if (N == 1)
    {
        document.write("1");
        return;
    }

    // No permutation is possible
    // satisfying the given condition
    if (N == 2 || N == 3)
    {
        document.write("-1");
        return;
    }

    // Maximum even and odd elements < N
    var even = -1, odd = -1;
    if (N % 2 == 0)
    {
        even = N;
        odd = N - 1;
    }
    else
    {
        odd = N;
        even = N - 1;
    }

    // Print var all odd elements in
    // decreasing order
    while (odd >= 1)
    {
        document.write(odd);
        document.write(" ");

        // Next element must be odd
        odd = odd - 2;
    }

    // Print var all even elements in
    // decreasing order
    while (even >= 2)
    {
        document.write(even);
        document.write(" ");

        // Next element must be even
        even = even - 2;
    }
}

// Driver code
var N = 5;

arrange(N);

// This code is contributed by umadevi9616

</script>

Output:

5 3 1 4 2

Time complexity: O(N), where N is the given integer.
Auxiliary space: O(1)