CSES Solutions - Permutations

A permutation of integers 1,2 ... N is called beautiful if there are no adjacent elements whose difference is 1. Given N, construct a beautiful permutation if such a permutation exists. If there are no solutions, print "NO SOLUTION".

Examples:

Input: N = 5
Output: 4 2 5 3 1
Explanation: No two adjacent elements have a difference of 1.

Input: N = 3
Output: NO SOLUTION
Explanation: No permutation of size 3 is possible.

Approach: To solve the problem, follow the below idea:

The idea is to construct a beautiful permutation by first outputting all the even numbers up to n and then all the odd numbers up to n. This approach ensures that no two adjacent elements have a difference of 1, as even numbers have a difference of at least 2 from each other, and the same holds for odd numbers. For n=2 and n=3, there is no possible solution.

Proof of Correctness:

The correctness of the code can be proven by examining the characteristics of even and odd numbers. The solution can be proved using 3 cases:-

1. Even-Even Adjacent Pair: For even numbers, the difference between consecutive elements is always 2 (e.g., 2, 4, 6, ...). Therefore, if we print all even numbers first, no two even numbers will be adjacent with a difference of 1.

2. Odd-Odd Adjacent Pair: For odd numbers, the difference between consecutive elements is also always 2 (e.g., 1, 3, 5, ...). If we print all odd numbers after the even numbers, no two odd numbers will be adjacent with a difference of 1.

3. Even-Odd Adjacent Pair: We output all even numbers first and then all odd numbers, so there is one even-odd adjacent pair which consists of the greatest even number less than or equal to n, and the first odd number will be 1. If the difference between these two numbers is greater than 1, then the permutation is valid, satisfying the conditions of a beautiful permutation. For n is greater than 3, the difference between the greatest even number less than or equal to n and the first odd number is always greater than 1, so our solution is correct. For n=2 or n=3, no permutation is possible.

Step-by-step algorithm:

• Check if n is 2 or 3. If so, it is not possible to construct a beautiful permutation, and "NO SOLUTION" is printed. Otherwise,
• Iterate through all even numbers from 2 to n with a step size of 2 and print them.
• Iterate through all odd numbers from 1 to n with a step size of 2 and print them.

Below is the implementation of above approach:

#include <bits/stdc++.h>

using namespace std;

// Function to construct a beautiful permutation
void permutation(int N)
{
    // Check if N is 2 or 3, as a beautiful permutation is
    // not possible for these cases
    if (N == 2 || N == 3) {
        cout << "NO SOLUTION\n";
        return;
    }

    // Output all even numbers first
    for (int i = 2; i <= N; i = i + 2) {
    
        // Print even numbers with a step of 2
        cout << i << " ";
    }

    // Output all odd numbers next
    for (int i = 1; i <= N; i = i + 2) {
    
        // Print odd numbers with a step of 2
        cout << i << " ";
    }
}

// Driver Code
int main()
{
    int N = 5;
    // Call the permutation function with input N
    permutation(N);
}

public class BeautifulPermutation {

    // Function to construct a beautiful permutation
    public static void permutation(int N) {
        // Check if N is 2 or 3, as a beautiful permutation is not possible for these cases
        if (N == 2 || N == 3) {
            System.out.println("NO SOLUTION");
            return;
        }

        // Output all even numbers first
        for (int i = 2; i <= N; i += 2) {
            System.out.print(i + " ");
        }

        // Output all odd numbers next
        for (int i = 1; i <= N; i += 2) {
            System.out.print(i + " ");
        }

        System.out.println();
    }

    public static void main(String[] args) {
        int N = 5;
        // Call the permutation function with input N
        permutation(N);
    }
}

def permutation(N):
    # Check if N is 2 or 3, as a beautiful permutation is not possible for these cases
    if N == 2 or N == 3:
        print("NO SOLUTION")
        return

    # Output all even numbers first
    for i in range(2, N + 1, 2):
        print(i)

    # Output all odd numbers next
    for i in range(1, N + 1, 2):
        print(i)


# Driver Code
if __name__ == "__main__":
    N = 5
    # Call the permutation function with input N
    permutation(N)

using System;

class BeautifulPermutation
{
    // Function to construct a beautiful permutation
    static void Permutation(int N)
    {
        // Check if N is 2 or 3, as a beautiful permutation is not possible for these cases
        if (N == 2 || N == 3)
        {
            Console.WriteLine("NO SOLUTION");
            return;
        }

        // Output all even numbers first
        for (int i = 2; i <= N; i += 2)
        {
            Console.Write(i + " ");
        }

        // Output all odd numbers next
        for (int i = 1; i <= N; i += 2)
        {
            Console.Write(i + " ");
        }

        Console.WriteLine();
    }

    public static void Main(string[] args)
    {
        int N = 5;
        // Call the permutation function with input N
        Permutation(N);
    }
}

function permutation(N) {
    // Check if N is 2 or 3, as a beautiful permutation is not possible for these cases
    if (N === 2 || N === 3) {
        console.log("NO SOLUTION");
        return;
    }

    let result = '';

    // Output all even numbers first
    for (let i = 2; i <= N; i += 2) {
        result += i + ' ';
    }

    // Output all odd numbers next
    for (let i = 1; i <= N; i += 2) {
        result += i + ' ';
    }

    console.log(result.trim());
}

// Driver Code
const N = 5;
// Call the permutation function with input N
permutation(N);

2 4 1 3 5 

Time Complexity: O(N), where N is the size of permutation we have to construct.
Auxiliary Space: O(1)