Minimum number of given operations required to be performed to reduce N to 0

Given an integer N, the task is to reduce N to 0 in the minimum number of operations using the following operations any number of times:

• Change the rightmost (0^th) bit in the binary representation of N.
• Change the i^th bit in the binary representation of N if the (i-1)^th bit is set to 1 and the (i-2)^th through 0^th bits are set to 0.

Examples:

Input: N = 3
Output: 2
Explanation: The binary representation of 3 is “11”.
“11″ ? “01” with the 2nd operation since the 0th bit is 1.
“01″ ? “00″ with the 1st operation.

Input: N = 6
Output: 4
Explanation: The binary representation of 6 is “110”.
“110” ? “010” with the 2nd operation since the 1st bit is 1 and 0th through 0th bits are 0.
“010” ? “011” with the 1st operation.
“011” ? “001” with the 2nd operation since the 0th bit is 1.
“001” ? “000” with the 1st operation.

Approach: The idea is based on the following observations:

• 1 ? 0 needs 1 operation.
  2 ? 0 i.e 10 ? 11 ? 01 ? 0 needs 3 operations.
  4 ? 0 i.e 100 ? 101 ? 111 ? 110 ? 010 ? 11 ? 01 ?0 needs 7 operations.
  Hence, it can be generalized for any power of 2 i.e, 2^k needs 2^(k+1) -1 operations.
• If a ? b requires k operation, b ? a also requires k operation.

Intermediate numbers from 2ⁿ to 2^(n-1) contain all numbers between 2ⁿ and 2^(n-1), which demonstrates that any given non-negative integer can be converted to 0. Let f(n) be a function to find the minimum number of operations required. The recurrence relation is: 
f(n) = f(2^k) – f(n xor 2^k), where k = next bit of most significant bit of n.

The idea is to use recursion. Follow the steps below to solve the problem:

• Create a recursive function that takes a number N as a parameter.
  • If the value of N is equal to 0, return 0.
  • Else, find the highest power of 2 less than or equal to N and store it in a variable X.
  • Store the value returned by recursively calling the function for (X^(X/2)^N) in a variable S.
  • Add the value of X to S and return it.

Below is the implementation of the above approach:

C++

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

// Function to find minimum
// operations required to convert
// N to 0
int minimumOneBitOperations(int n, int res = 0)
{
    // Base Case
    if (n == 0)
        return res;

    // Store the highest power of 2
    // less than or equal to n
    int b = 1;
    while ((b << 1) <= n)
        b = b << 1;

    // Return the result
    return minimumOneBitOperations(
        (b >> 1) ^ b ^ n, res + b);
}

// Driver Code
int main()
{
    // Given Input
    int N = 6;

    // Function call
    cout << minimumOneBitOperations(N);

    return 0;
}

Java

// Java program for the above approach
class GFG {

    // Function to find minimum
    // operations required to convert
    // N to 0
    public static int minimumOneBitOperations(int n, int res)
    {

        // Base Case
        if (n == 0)
            return res;

        // Store the highest power of 2
        // less than or equal to n
        int b = 1;
        while ((b << 1) <= n)
            b = b << 1;

        // Return the result
        return minimumOneBitOperations((b >> 1) ^ b ^ n, res + b);
    }

    // Driver Code
    public static void main(String args[])
    {

        // Given Input
        int N = 6;

        // Function call
        System.out.println(minimumOneBitOperations(N, 0));
    }
}

// This code is contributed by gfgking.

Python3

# Python program for the above approach

# Function to find minimum
# operations required to convert
# N to 0
def minimumOneBitOperations(n, res):

  # Base case
    if n == 0:
        return res

     # Store the highest power of 2
    # less than or equal to n
    b = 1
    while (b << 1) <= n:
        b = b << 1

    # Return the result
    return minimumOneBitOperations((b >> 1) ^ b ^ n, res + b)

# Driver code
N = 6
print(minimumOneBitOperations(N, 0))

# This code is contributed by Parth Manchanda

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to find minimum
// operations required to convert
// N to 0
static int minimumOneBitOperations(int n, int res)
{
    // Base Case
    if (n == 0)
        return res;

    // Store the highest power of 2
    // less than or equal to n
    int b = 1;
    while ((b << 1) <= n)
        b = b << 1;

    // Return the result
    return minimumOneBitOperations(
        (b >> 1) ^ b ^ n, res + b);
}

// Driver Code
public static void Main()
{
    // Given Input
    int N = 6;

    // Function call
    Console.Write(minimumOneBitOperations(N,0));
}
}

// This code is contributed by SURENDRA_GANGWAR.

Javascript

<script>
    // Javascript program for the above approach


// Function to find minimum
// operations required to convert
// N to 0
function minimumOneBitOperations( n,  res = 0)
{
    // Base Case
    if (n == 0)
        return res;

    // Store the highest power of 2
    // less than or equal to n
    let b = 1;
    while ((b << 1) <= n)
        b = b << 1;

    // Return the result
    return minimumOneBitOperations((b >> 1) ^ b ^ n, res + b);
}

// Driver Code

    // Given Input
    let N = 6;

    // Function call
    document.write(minimumOneBitOperations(N));

// This code is contributed by
// Potta Lokesh

    </script>

Output:

4

Time Complexity: O(log(N))
Auxiliary Space: O(1)