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Find a permutation of 2N numbers such that the result of given expression is exactly 2K

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  • Last Updated : 23 Jun, 2022
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Given two integers N and K, the task is to find a permutation of first 2*N natural numbers such that the following equation is satisfied. 
\sum\limits_{i=1}^N |A_{2i-1}-A_{2i}| - |\sum\limits_{i=1}^N A_{2i-1}-A_{2i}|=2K
Note: The value of K will always be less than or equal to N.
Examples: 
 

Input : N = 1,  K = 0
Output : 1 2
The result of the above expression will be:
|1-2|-|1-2| =0

Input : N = 2,  K = 1 
Output : 2 1 3 4
The result of the above expression will be:
(|2-1|+|3-4|)-(|2-1+3-4|) = 2

 

Approach:
Consider the sorted permutation: 
 

1, 2, 3, 4, 5, 6.... 

The result of the expression will come out to be exactly 0. If we swap any 2 indices 2i-1 and 2i, the result will increase by exactly 2. So we need to make K such swaps. 
Below is the implementation of the above approach: 
 

C++




// C++ program to find the required permutation
// of first 2*N natural numbers
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the required permutation
// of first 2*N natural numbers
void printPermutation(int n, int k)
{
    // Iterate in blocks of 2
    for (int i = 1; i <= n; i++) {
        int x = 2 * i - 1;
        int y = 2 * i;
 
        // We need more increments, so print in reverse order
        if (i <= k)
            cout << y << " " << x << " ";
 
        // We have enough increments, so print in same order
        else
            cout << x << " " << y << " ";
    }
}
 
// Driver Code
int main()
{
    int n = 2, k = 1;
 
    printPermutation(n, k);
 
    return 0;
}

Java




// Java program to find the
// required permutation
// of first 2*N natural numbers
class GFG
{
         
    // Function to find the required permutation
    // of first 2*N natural numbers
    static void printPermutation(int n, int k)
    {
        // Iterate in blocks of 2
        for (int i = 1; i <= n; i++)
        {
            int x = 2 * i - 1;
            int y = 2 * i;
         
        // We need more increments,
        // so print in reverse order
        if (i <= k)
            System.out.print(y + " " + x + " ");
 
        // We have enough increments,
        // so print in same order
        else
            System.out.print(x + " " + y + " ");
        }
    }
         
    // Driver code
    public static void main(String []args)
    {
        int n = 2, k = 1;
        printPermutation(n, k);
    }
}
 
// This code is contributed by Ita_c.

Python3




# Python3 program to find the required
# permutation of first 2*N natural numbers
 
# Function to find the required permutation
# of first 2*N natural numbers
def printPermutation(n, k) :
     
    # Iterate in blocks of 2
    for i in range(1, n + 1) :
        x = 2 * i - 1;
        y = 2 * i;
 
        # We need more increments,
        # so print in reverse order
        if (i <= k) :
            print(y, x, end = " ");
 
        # We have enough increments,
        # so print in same order
        else :
            print(x, y, end = " ");
 
# Driver Code
if __name__ == "__main__" :
    n = 2; k = 1;
 
    printPermutation(n, k);
     
# This code is contributed by Ryuga

C#




using System;
 
// C# program to find the
// required permutation
// of first 2*N natural numbers
     
class GFG
{
         
    // Function to find the required permutation
    // of first 2*N natural numbers
    static void printPermutation(int n, int k)
    {
        // Iterate in blocks of 2
        for (int i = 1; i <= n; i++)
        {
            int x = 2 * i - 1;
            int y = 2 * i;
         
        // We need more increments,
        // so print in reverse order
        if (i <= k)
            Console.Write(y + " " + x + " ");
 
        // We have enough increments,
        // so print in same order
        else
            Console.Write(x + " " + y + " ");
        }
    }
         
    // Driver code
    public static void Main()
    {
        int n = 2, k = 1;
        printPermutation(n, k);
    }
}
 
// This code is contributed by
// shashank_sharma

PHP




<?php
// PHP program to find the required
// permutation of first 2*N natural numbers
 
// Function to find the required permutation
// of first 2*N natural numbers
function printPermutation($n, $k)
{
    // Iterate in blocks of 2
    for ($i = 1; $i <= $n; $i++)
    {
        $x = 2 * $i - 1;
        $y = 2 * $i;
 
        // We need more increments, so print
        // in reverse order
        if ($i <= $k)
            echo $y . " " . $x . " ";
 
        // We have enough increments,
        // so print in same order
        else
            echo $x . " " . $y . " ";
    }
}
 
// Driver Code
$n = 2;
$k = 1;
 
printPermutation($n, $k);
 
// This code is contributed by chandan_jnu
?>

Javascript




<script>
 
// javascript program to find the
// required permutation
// of first 2*N natural numbers
     
  
         
    // Function to find the required permutation
    // of first 2*N natural numbers
    function printPermutation( n,  k)
    {
        // Iterate in blocks of 2
        for (var i = 1; i <= n; i++)
        {
            var x = 2 * i - 1;
            var y = 2 * i;
         
        // We need more increments,
        // so print in reverse order
        if (i <= k)
            document.write(y + " " + x + " ");
 
        // We have enough increments,
        // so print in same order
        else
            document.write(x + " " + y + " ");
        }
    }
         
    // Driver code
  
        var n = 2, k = 1;
        printPermutation(n, k);
         
        // This code is contributed by bunnyram19.
         
</script>

Output: 

2 1 3 4

 

Time Complexity: O(N), since there runs a loop from 1 to n.
Auxiliary Space: O(1), since no extra space has been taken.
 


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