Generate permutation of 1 to N such that absolute difference of consecutive numbers give K distinct integers

Given two integers N and K where K < N, the task is to generate a permutation of integers from 1 to N such that the absolute difference of all the consecutive integers give exactly K distinct integers.

Examples:

Input: N = 3, K = 2
Output: 1 3 2
|1 – 3| = 2 and |3 – 2| = 1 which gives 2 distinct integers (2 and 1)

Input: N = 5, K = 4
Output: 1 5 2 4 3
|1 – 5| = 4, |5 – 2| = 3, |2 – 4| = 2 and |4 – 3| = 1 gives 4 distinct integers i.e. 4, 3, 2 and 1



Approach: The problem can be easily solved by simple observation. At the odd indices place increasing sequence 1, 2, 3, … and at the even indices place the decreasing sequence N, N-1, N-2, … and so on.

For N = 10, a permutation with distinct integers for consecutive absolute difference can be 1 10 2 9 3 8 4 7 5 6. The consecutive absolute difference gives integers 9, 8, 7 and so on.
So, first print K integers of such a sequence then make the rest of the differences equal to 1. The code is quite self explanatory.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to to generate a permutation of integers
// from 1 to N such that the absolute difference of
// all the two consecutive integers give K distinct integers
void printPermutation(int N, int K)
{
    // To store the permutation
    vector<int> res;
  
    int l = 1, r = N, flag = 0;
  
    for (int i = 0; i < K; i++) {
  
        if (!flag) {
  
            // For sequence 1 2 3...
            res.push_back(l);
            l++;
        }
        else {
  
            // For sequence N, N-1, N-2...
            res.push_back(r);
            r--;
        }
  
        // Flag is used to alternate between
        // the above if else statements
        flag ^= 1;
    }
  
    // Taking integers with difference 1
  
    // If last element added was r + 1
    if (!flag) {
        for (int i = r; i >= l; i--)
            res.push_back(i);
    }
  
    // If last element added was l - 1
    else {
        for (int i = l; i <= r; i++)
            res.push_back(i);
    }
  
    // Print the permutation
    for (auto i : res)
        cout << i << " ";
}
  
// Driver code
int main()
{
    int N = 10, K = 4;
  
    printPermutation(N, K);
  
    return 0;
}

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Java

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// Java implementation of the above approach
import java.util.Vector;
  
class GFG 
{
    // Function to to generate a permutation 
    // of integers from 1 to N such that the 
    // absolute difference of all the two 
    // consecutive integers give K distinct integers
    static void printPermutation(int N, int K) 
    {
        // To store the permutation
        Vector<Integer> res = new Vector<>();
  
        int l = 1, r = N, flag = 0;
  
        for (int i = 0; i < K; i++)
        {
            if (flag == 0
            {
                // For sequence 1 2 3...
                res.add(l);
                l++;
            
            else 
            {
                // For sequence N, N-1, N-2...
                res.add(r);
                r--;
            }
  
            // Flag is used to alternate between
            // the above if else statements
            flag ^= 1;
        }
  
        // Taking integers with difference 1
        // If last element added was r + 1
        if (flag != 1
        {
            for (int i = r; i >= l; i--) 
            {
                res.add(i);
            }
        
        // If last element added was l - 1
        else 
        {
            for (int i = l; i <= r; i++) 
            {
                res.add(i);
            }
        }
  
        // Print the permutation
        for (Integer i : res) 
        {
            System.out.print(i + " ");
        }
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int N = 10, K = 4;
        printPermutation(N, K);
          
    }
}
  
// This code is contributed by
// 29AjayKumar 

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Python3

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# Python 3 implementation of the approach
  
# Function to to generate a permutation 
# of integers from 1 to N such that the 
# absolute difference of all the two 
# consecutive integers give K distinct
# integers
def printPermutation(N, K):
  
    # To store the permutation
    res = list();
  
    l, r, flag = 1, N, 0
  
    for i in range(K):
  
        if flag == False:
              
            # For sequence 1 2 3...
            res.append(l)
            l += 1
      
        else:
              
            # For sequence N, N-1, N-2...
            res.append(r);
            r -= 1;
  
    # Flag is used to alternate between
    # the above if else statements
        flag = flag ^ 1;
  
    # Taking integers with difference 1
  
    # If last element added was r + 1
    if flag == False:
        for i in range(r, 2, -1):
            res.append(i)
  
    # If last element added was l - 1
    else:
        for i in range(l, r):
            res.append(i)
  
    # Print the permutation
    for i in res:
        print(i, end = " ")
  
# Driver code
N, K = 10, 4
printPermutation(N, K)
  
# This code is contributed by
# Mohit Kumar 29

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C#

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// C# implementation of the above approach
using System;
using System.Collections;
  
class GFG 
{
    // Function to to generate a permutation 
    // of integers from 1 to N such that the 
    // absolute difference of all the two 
    // consecutive integers give K distinct integers
    static void printPermutation(int N, int K) 
    {
          
        // To store the permutation
        ArrayList res = new ArrayList();
  
        int l = 1, r = N, flag = 0;
        for (int i = 0; i < K; i++)
        {
            if (flag == 0) 
            {
                // For sequence 1 2 3...
                res.Add(l);
                l++;
            
            else
            {
                // For sequence N, N-1, N-2...
                res.Add(r);
                r--;
            }
  
            // Flag is used to alternate between
            // the above if else statements
            flag ^= 1;
        }
  
        // Taking integers with difference 1
        // If last element added was r + 1
        if (flag != 1) 
        {
            for (int i = r; i >= l; i--) 
            {
                res.Add(i);
            }
        
          
        // If last element added was l - 1
        else
        {
            for (int i = l; i <= r; i++) 
            {
                res.Add(i);
            }
        }
  
        // Print the permutation
        foreach (int i in res) 
        {
            Console.Write(i + " ");
        }
    }
  
    // Driver code
    public static void Main()
    {
        int N = 10, K = 4;
        printPermutation(N, K);     
    }
}
  
// This code is contributed by PrinciRaj1992

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PHP

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<?php
// PHP implementation of the approach 
  
// Function to to generate a permutation 
// of integers from 1 to N such that the 
// absolute difference of all the two 
// consecutive integers give K distinct 
// integers 
function printPermutation($N, $K
      
    // To store the permutation 
    $res = array();
  
    $l = 1;
    $r = $N;
    $flag = 0; 
  
    for ($i = 0; $i < $K; $i++) 
    
        if (!$flag
        
  
            // For sequence 1 2 3... 
            array_push($res, $l); 
            $l++; 
        
        else 
        
  
            // For sequence N, N-1, N-2... 
            array_push($res, $r); 
            $r--; 
        
  
        // Flag is used to alternate between 
        // the above if else statements 
        $flag ^= 1; 
    
  
    // Taking integers with difference 1 
  
    // If last element added was r + 1 
    if (!$flag
    
        for ($i = $r; $i >= $l; $i--) 
            array_push($res, $i); 
    
  
    // If last element added was l - 1 
    else 
    
        for ($i = l; $i <= $r; $i++) 
            array_push($res, $i); 
    
  
    // Print the permutation 
    for($i = 0; $i < sizeof($res); $i++)
        echo $res[$i], " ";
  
// Driver code 
$N = 10;
$K = 4; 
  
printPermutation($N, $K); 
  
// This code is contributed by Ryuga
?>

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Output:

1 10 2 9 8 7 6 5 4 3


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