Generate a permutation of first N natural numbers having count of unique adjacent differences equal to K | Set 2

Last Updated : 24 Feb, 2022

Given two positive integers N and K, the task is to construct a permutation of the first N natural numbers such that all possible absolute differences between adjacent elements are K.

Examples:

Input: N = 3, K = 1
Output: 1 2 3
Explanation: Considering the permutation {1, 2, 3}, all possible unique absolute difference of adjacent elements is {1}. Since the count is 1(= K), print the sequence {1, 2, 3} as the resultant permutation.

Input: N = 3, K = 2
Output: 1 3 2

The naive approach and the two-pointer approach of this problem are already discussed here. This article discusses a different approach deque.

Approach: It is easy to see that, answers for all values of K between [1, N-1] can be generated. For any K outside this range, there exists no answer. To solve the problem maintain a double-ended queue for all the current elements and a vector to store the sequence. Also, maintain a boolean value that will help to determine to pop the front or back element. Iterate the remaining element and if K is greater than 1 then push the element according to the boolean value and decrease K by 1. Flip the boolean value so that all remaining differences will have value 1. Follow the steps below to solve the problem:

If K is greater than equal to N or less than 1, then print -1 and return as no such sequence exists.
Initialize a deque dq[] to maintain the order of elements.
Iterate over the range [2, N] using the variable i and push i into the back of deque dq[].
Initialize a boolean variable front as true.
Initialize a vector ans[] to store the answer and push 1 into the vector ans[].
If K is greater than 1, then subtract 1 from K and xor the value of front with 1.
Iterate over the range [2, N] using the variable i and check:
- If the front is 1, then pop the front of the deque dq[] and push it in the vector ans[] otherwise, pop the back of the deque dq[] and push it in the vector ans[]. Also, if K is greater then, then subtract 1 from K and xor the value of front with 1.
Traverse the array ans[] and print its elements.

Below is the implementation of the above approach.

C++

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

// Function to calculate the required array
void K_ValuesArray(int N, int K)
{

    // Check for base cases
    if (K < 1 || K >= N) {
        cout << -1;
        return;
    }

    // Maintain a deque to store the
    // elements from [1, N];
    deque<int> dq;
    for (int i = 2; i <= N; i++) {
        dq.push_back(i);
    }

    // Maintain a boolean value which will
    // tell from where to pop the element
    bool front = true;

    // Create a vector to store the answer
    vector<int> ans;

    // Push 1 in the answer initially
    ans.push_back(1);

    // Push the remaining elements
    if (K > 1) {
        front ^= 1;
        K--;
    }

    // Iterate over the range
    for (int i = 2; i <= N; i++) {
        if (front) {
            int val = dq.front();
            dq.pop_front();

            // Push this value in
            // the ans vector
            ans.push_back(val);
            if (K > 1) {
                K--;

                // Flip the boolean
                // value
                front ^= 1;
            }
        }
        else {
            int val = dq.back();
            dq.pop_back();

            // Push value in ans vector
            ans.push_back(val);
            if (K > 1) {
                K--;

                // Flip boolean value
                front ^= 1;
            }
        }
    }

    // Print Answer
    for (int i = 0; i < N; i++) {
        cout << ans[i] << " ";
    }
}

// Driver Code
int main()
{
    int N = 7, K = 1;
    K_ValuesArray(N, K);

    return 0;
}

Java

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

// Function to calculate the required array
static void K_ValuesArray(int N, int K)
{

    // Check for base cases
    if (K < 1 || K >= N) {
        System.out.print(-1);
        return;
    }

    // Maintain a deque to store the
    // elements from [1, N];
    Deque<Integer> dq = new LinkedList<Integer>();
    for (int i = 2; i <= N; i++) {
        dq.add(i);
    }

    // Maintain a boolean value which will
    // tell from where to pop the element
    boolean front = true;

    // Create a vector to store the answer
    Vector<Integer> ans = new Vector<Integer>();

    // Push 1 in the answer initially
    ans.add(1);

    // Push the remaining elements
    if (K > 1) {
        front ^=true;
        K--;
    }

    // Iterate over the range
    for (int i = 2; i <= N; i++) {
        if (front) {
            int val = dq.peek();
            dq.removeFirst();

            // Push this value in
            // the ans vector
            ans.add(val);
            if (K > 1) {
                K--;

                // Flip the boolean
                // value
                front ^=true;
            }
        }
        else {
            int val = dq.getLast();
            dq.removeLast();

            // Push value in ans vector
            ans.add(val);
            if (K > 1) {
                K--;

                // Flip boolean value
                front ^=true;
            }
        }
    }

    // Print Answer
    for (int i = 0; i < N; i++) {
        System.out.print(ans.get(i)+ " ");
    }
}

// Driver Code
public static void main(String[] args)
{
    int N = 7, K = 1;
    K_ValuesArray(N, K);

}
}

// This code is contributed by 29AjayKumar

Python3

# python Program for the above approach
from collections import deque

# Function to calculate the required array
def K_ValuesArray(N, K):

    # Check for base cases
    if (K < 1 or K >= N):
        print("-1")
        return

    # Maintain a deque to store the
    # elements from [1, N];
    dq = deque()
    for i in range(2, N + 1):
        dq.append(i)

    # Maintain a boolean value which will
    # tell from where to pop the element
    front = True

    # Create a vector to store the answer
    ans = []

    # Push 1 in the answer initially
    ans.append(1)

    # Push the remaining elements
    if (K > 1):
        front ^= 1
        K -= 1

    # Iterate over the range
    for i in range(2, N+1):
        if (front):
            val = dq.popleft()

            # Push this value in
            # the ans vector
            ans.append(val)
            if (K > 1):
                K -= 1

                # Flip the boolean
                # value
                front ^= 1

        else:
            val = dq.pop()

            # Push value in ans vector
            ans.append(val)
            if (K > 1):
                K -= 1

                # Flip boolean value
                front ^= 1

    # Print Answer
    for i in range(0, N):
        print(ans[i], end=" ")

# Driver Code
if __name__ == "__main__":

    N = 7
    K = 1
    K_ValuesArray(N, K)

    # This code is contributed by rakeshsahni

// C# Program for the above approach

using System;
using System.Collections.Generic;
class GFG
{

    // Function to calculate the required array
    static void K_ValuesArray(int N, int K)
    {

        // Check for base cases
        if (K < 1 || K >= N)
        {
            Console.Write(-1);
            return;
        }

        // Maintain a deque to store the
        // elements from [1, N];
        LinkedList<int> dq = new LinkedList<int>();
        for (int i = 2; i <= N; i++)
        {
            dq.AddLast(i);
        }

        // Maintain a boolean value which will
        // tell from where to pop the element
        bool front = true;

        // Create a vector to store the answer
        List<int> ans = new List<int>();

        // Push 1 in the answer initially
        ans.Add(1);

        // Push the remaining elements
        if (K > 1)
        {
            front ^= true;
            K--;
        }

        // Iterate over the range
        for (int i = 2; i <= N; i++)
        {
            if (front)
            {
                int val = dq.First.Value;
                dq.RemoveFirst();

                // Push this value in
                // the ans vector
                ans.Add(val);
                if (K > 1)
                {
                    K--;

                    // Flip the boolean
                    // value
                    front ^= true;
                }
            }
            else
            {
                int val = dq.Last.Value;
                dq.RemoveLast();

                // Push value in ans vector
                ans.Add(val);
                if (K > 1)
                {
                    K--;

                    // Flip boolean value
                    front ^= true;
                }
            }
        }

        // Print Answer
        for (int i = 0; i < N; i++)
        {
            Console.Write(ans[i] + " ");
        }
    }

    // Driver Code
    public static void Main()
    {
        int N = 7, K = 1;
        K_ValuesArray(N, K);

    }
}

// This code is contributed by Saurabh Jaiswal

JavaScript

<script>
// Javascript Program for the above approach

// Function to calculate the required array
function K_ValuesArray(N, K) {

    // Check for base cases
    if (K < 1 || K >= N) {
        document.write(-1);
        return;
    }

    // Maintain a deque to store the
    // elements from [1, N];
    let dq = new Array();
    for (let i = 2; i <= N; i++) {
        dq.push(i);
    }

    // Maintain a boolean value which will
    // tell from where to pop the element
    let front = true;

    // Create a vector to store the answer
    let ans = new Array();

    // Push 1 in the answer initially
    ans.push(1);

    // Push the remaining elements
    if (K > 1) {
        front ^= true;
        K--;
    }

    // Iterate over the range
    for (let i = 2; i <= N; i++) {
        if (front) {
            let val = dq[0];
            dq.shift();

            // Push this value in
            // the ans vector
            ans.push(val);
            if (K > 1) {
                K--;

                // Flip the boolean
                // value
                front ^= true;
            }
        }
        else {
            let val = dq.Last.Value;
            dq.pop();

            // Push value in ans vector
            ans.push(val);
            if (K > 1) {
                K--;

                // Flip boolean value
                front ^= true;
            }
        }
    }

    // Print Answer
    for (let i = 0; i < N; i++) {
        document.write(ans[i] + " ");
    }
}

// Driver Code
let N = 7, K = 1;
K_ValuesArray(N, K);

// This code is contributed by Saurabh Jaiswal
</script>

Output

1 2 3 4 5 6 7

Time Complexity: O(N)
Auxiliary Space: O(N)

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

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Generate a permutation of first N natural numbers having count of unique adjacent differences equal to K | Set 2

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