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Count of total bits toggled/flipped in binary representation of 0 to N

  • Difficulty Level : Expert
  • Last Updated : 31 Mar, 2021

Given an integer N, the task is to find the total number of bits toggled to obtain all numbers from 0 to N sequentially.

Examples:

Input: N = 5 
Output:
Explanation: 
Let’s represent numbers from 0 to 5 in binary: 
000 -> 001 : 1 bit toggled 
001 -> 010 : 2 bits toggled 
010 -> 011 : 1 bit toggled 
011 -> 100 : 3 bits toggled 
100 -> 101 : 1 bit toggled 
Hence, 8 bits toggled

Input: N = 1 
Output: 1

Approach: 
Follow the steps below to solve the problems:



  • The following observations need to be made to solve the problem:

 
 The rightmost bit toggles every time. 
(000) -> (001) -> (010) -> (011
So, the contribution of this bit to the count will be N
The next bit toggles after every 2 numbers. 
(000) -> (001) -> (010) -> (011) -> (100) 
Hence, the contribution of this bit to the count will be N/2.

 

  • Thus, we can conclude that the i th least significant bit will contribute N/(2 i) to the count of toggles.
  • Hence, the sum of N/(2i) where i is in the range [0, log2N], gives the required answer.
  • Hence, initialize a variable ans. Add N to ans and update N to N/2. Repeat this process until N becomes 0, to get the final result.

Below is the implementation of the above approach:

C++




// C++ program to count
// the number of toggles
// required to generate
// all numbers from 0 to N
 
#include <bits/stdc++.h>
using namespace std;
 
typedef long long int ll;
 
// Function to count and print
// the required number of
// toggles
void solve(ll N)
{
    // Store the count
    // of toggles
    ll ans = 0;
 
    while (N != 0) {
 
        // Add the contribution
        // of the current LSB
        ans += N;
 
        // Update N
        N /= 2;
    }
    // Print the result
    cout << ans << endl;
}
// Driver code
int main()
{
    ll N = 5;
    solve(N);
    return 0;
}

Java




// Java program to count the
// number of toggles required
// to generate all numbers
// from 0 to N
class GFG{
 
// Function to count and print
// the required number of
// toggles
static void solve(int N)
{
     
    // Store the count
    // of toggles
    int ans = 0;
 
    while (N != 0)
    {
         
        // Add the contribution
        // of the current LSB
        ans += N;
 
        // Update N
        N /= 2;
    }
     
    // Print the result
    System.out.println(ans);
}
 
// Driver code
public static void main(String []args)
{
    int N = 5;
     
    solve(N);
}
}
 
// This code is contributed by Ritik Bansal

Python3




# Python3 program to count
# the number of toggles
# required to generate
# all numbers from 0 to N
 
# Function to count and pr
# the required number of
# toggles
def solve(N):
     
    # Store the count
    # of toggles
    ans = 0
 
    while (N != 0):
 
        # Add the contribution
        # of the current LSB
        ans += N
 
        # Update N
        N //= 2
     
    # Print the result
    print(ans)
 
# Driver code
N = 5
 
solve(N)
 
# This code is contributed by code_hunt

C#




// C# program to count the
// number of toggles required
// to generate all numbers
// from 0 to N
using System;
 
class GFG{
 
// Function to count and print
// the required number of
// toggles
static void solve(int N)
{
     
    // Store the count
    // of toggles
    int ans = 0;
 
    while (N != 0)
    {
         
        // Add the contribution
        // of the current LSB
        ans += N;
 
        // Update N
        N /= 2;
    }
     
    // Print the result
    Console.Write(ans);
}
 
// Driver code
public static void Main(string []args)
{
    int N = 5;
     
    solve(N);
}
}
 
// This code is contributed by rock_cool

Javascript




<script>
 
// Javascript program to count the
// number of toggles required to
// generate all numbers from 0 to N
 
// Function to count and print
// the required number of
// toggles
function solve(N)
{
     
    // Store the count
    // of toggles
    let ans = 0;
 
    while (N != 0)
    {
         
        // Add the contribution
        // of the current LSB
        ans += N;
 
        // Update N
        N = parseInt(N / 2, 10);
    }
     
    // Print the result
    document.write(ans);
}
     
// Driver code   
let N = 5;
 
solve(N);
     
// This code is contributed by divyeshrabadiya07
 
</script>

Output:

8

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

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