Represent n as the sum of exactly k powers of two | Set 2

Given two integers n and k, the task is to find whether it is possible to represent n as the sum of exactly k powers of 2. If possible then print k positive integers such that they are powers of 2 and their sum is exactly equal to n else print Impossible.

Examples:

Input: n = 9, k = 4 
Output: 1 2 2 4 
1, 2 and 4 are all powers of 2 and 1 + 2 + 2 + 4 = 9.

Input: n = 3, k = 7 
Output: Impossible 
It is impossible since 3 cannot be represented as sum of 7 numbers which are powers of 2. 

We have discussed one approach to solve this problem in Find k numbers which are powers of 2 and have sum N. In this post, a different approach is being discussed.

Approach:

• Create an array arr[] of size k with all elements initialized to 1 and create a variable sum = k.
• Now starting from the last element of arr[]
  • If sum + arr[i] ? n then update sum = sum + arr[i] and arr[i] = arr[i] * 2.
  • Else skip the current element.
• If sum = n then the contents of arr[] are the required elements.
• Else it is impossible to represent n as exactly k powers of 2.

Below is the implementation of the above approach: 

C++

// C++ implementation of the above approach
#include <iostream>
using namespace std;

// Function to print k numbers which are powers of two
// and whose sum is equal to n
void FindAllElements(int n, int k)
{
    // Initialising the sum with k
    int sum = k;

    // Initialising an array A with k elements
    // and filling all elements with 1
    int A[k];
    fill(A, A + k, 1);

    for (int i = k - 1; i >= 0; --i) {

        // Iterating A[] from k-1 to 0
        while (sum + A[i] <= n) {

            // Update sum and A[i]
            // till sum + A[i] is less than equal to n
            sum += A[i];
            A[i] *= 2;
        }
    }

    // Impossible to find the combination
    if (sum != n) {
        cout << "Impossible";
    }

    // Possible solution is stored in A[]
    else {
        for (int i = 0; i < k; ++i)
            cout << A[i] << ' ';
    }
}

// Driver code
int main()
{
    int n = 12;
    int k = 6;

    FindAllElements(n, k);

    return 0;
}

Java

// Java implementation of the above approach
import java.util.Arrays;

public class GfG {

    // Function to print k numbers which are powers of two
    // and whose sum is equal to n
    public static void FindAllElements(int n, int k)
    {
        // Initialising the sum with k
        int sum = k;

        // Initialising an array A with k elements
        // and filling all elements with 1
        int[] A = new int[k];
        Arrays.fill(A, 0, k, 1);

        for (int i = k - 1; i >= 0; --i) {

            // Iterating A[] from k-1 to 0
            while (sum + A[i] <= n) {

                // Update sum and A[i]
                // till sum + A[i] is less than equal to n
                sum += A[i];
                A[i] *= 2;
            }
        }

        // Impossible to find the combination
        if (sum != n) {
            System.out.print("Impossible");
        }

        // Possible solution is stored in A[]
        else {
            for (int i = 0; i < k; ++i)
                System.out.print(A[i] + " ");
        }
    }

    public static void main(String []args){

        int n = 12;
        int k = 6;

        FindAllElements(n, k);
    }
}

// This code is contributed by Rituraj Jain

Python3

# Python 3 implementation of the above approach

# Function to print k numbers which are
# powers of two and whose sum is equal to n
def FindAllElements(n, k):

    # Initialising the sum with k
    sum = k

    # Initialising an array A with k elements
    # and filling all elements with 1
    A = [1 for i in range(k)]
    i = k - 1
    while(i >= 0):

        # Iterating A[] from k-1 to 0
        while (sum + A[i] <= n):

            # Update sum and A[i] till
            # sum + A[i] is less than equal to n
            sum += A[i]
            A[i] *= 2
        i -= 1

    # Impossible to find the combination
    if (sum != n):
        print("Impossible")

    # Possible solution is stored in A[]
    else:
        for i in range(0, k, 1):
            print(A[i], end = ' ')

# Driver code
if __name__ == '__main__':
    n = 12
    k = 6

    FindAllElements(n, k)

# This code is contributed by
# Surendra_Gangwar

C#

// C# implementation of the above approach
using System;

class GfG
{

    // Function to print k numbers
    // which are powers of two
    // and whose sum is equal to n
    public static void FindAllElements(int n, int k)
    {
        // Initialising the sum with k
        int sum = k;

        // Initialising an array A with k elements
        // and filling all elements with 1
        int[] A = new int[k];
        for(int i = 0; i < k; i++)
            A[i] = 1;

        for (int i = k - 1; i >= 0; --i)
        {

            // Iterating A[] from k-1 to 0
            while (sum + A[i] <= n)
            {

                // Update sum and A[i]
                // till sum + A[i] is less than equal to n
                sum += A[i];
                A[i] *= 2;
            }
        }

        // Impossible to find the combination
        if (sum != n)
        {
            Console.Write("Impossible");
        }

        // Possible solution is stored in A[]
        else
        {
            for (int i = 0; i < k; ++i)
                Console.Write(A[i] + " ");
        }
    }

    // Driver code
    public static void Main(String []args)
    {

        int n = 12;
        int k = 6;

        FindAllElements(n, k);
    }
}

// This code contributed by Rajput-Ji

PHP

<?php
// PHP implementation of the above approach

// Function to print k numbers which are
// powers of two and whose sum is equal to n
function FindAllElements($n, $k)
{
    // Initialising the sum with k
    $sum = $k;

    // Initialising an array A with k elements
    // and filling all elements with 1
    $A = array_fill(0, $k, 1) ;


    for ($i = $k - 1; $i >= 0; --$i)
    {

        // Iterating A[] from k-1 to 0
        while ($sum + $A[$i] <= $n)
        {

            // Update sum and A[i] till
            // sum + A[i] is less than equal to n
            $sum += $A[$i];
            $A[$i] *= 2;
        }
    }

    // Impossible to find the combination
    if ($sum != $n)
    {
        echo"Impossible";
    }

    // Possible solution is stored in A[]
    else
    {
        for ($i = 0; $i < $k; ++$i)
            echo $A[$i], ' ';
    }
}

// Driver code
$n = 12;
$k = 6;

FindAllElements($n, $k);

// This code is contributed by Ryuga
?>

Javascript

<script>

// Javascript implementation of the above approach


// Function to print k numbers which are powers of two
// and whose sum is equal to n
function FindAllElements( n, k)
{
    // Initialising the sum with k
    let sum = k;

    // Initialising an array A with k elements
    // and filling all elements with 1
    let A = new Array(k).fill(1);


    for (let i = k - 1; i >= 0; --i) {

        // Iterating A[] from k-1 to 0
        while (sum + A[i] <= n) {

            // Update sum and A[i]
            // till sum + A[i] is less than equal to n
            sum += A[i];
            A[i] *= 2;
        }
    }

    // Impossible to find the combination
    if (sum != n) {
        document.write("Impossible");
    }

    // Possible solution is stored in A[]
    else {
        for (let i = 0; i < k; ++i)
            document.write(A[i] + " ");
    }
}


// Driver Code

let n = 12;
let k = 6;

FindAllElements(n, k);

</script>

Output

1 1 1 1 4 4 

Complexity Analysis:

• Time Complexity: O(k*log₂(n-k))
• Auxiliary Space: O(k), since k extra space has been taken.