Bits manipulation (Important tactics)

Prerequisites: Bitwise operators in C, Bitwise Hacks for Competitive Programming, Bit Tricks for Competitive Programming

Table of Contents

• Compute XOR from 1 to n (direct method)
• Count of numbers (x) smaller than or equal to n such that n+x = n^x
• How to know if a number is a power of 2?
• Find XOR of all subsets of a set
• Find the number of leading, trailing zeroes and number of 1’s
• Convert binary code directly into an integer in C++
• The Quickest way to swap two numbers
• Simple approach to flip the bits of a number
• Finding the most significant set bit (MSB)
• Check if a number has bits in an alternate pattern

1. Compute XOR from 1 to n (direct method):

The  problem can be solved based on the following observations:

Say x = n%4. The XOR value depends on the value if x. If

• x = 0, then the answer is n.
• x = 1, then answer is 1.
• x = 2, then answer is n+1.
• x = 3, then answer is 0.

Below is the implementation of the above approach.

CPP

// Direct XOR of all numbers from 1 to n
int computeXOR(int n)
{
    if (n % 4 == 0)
        return n;
    if (n % 4 == 1)
        return 1;
    if (n % 4 == 2)
        return n + 1;
    else
        return 0;
}

Java

/*package whatever //do not write package name here */
import java.io.*;

class GFG
{

  // Direct XOR of all numbers from 1 to n
  public static int computeXOR(int n)
  {
    if (n % 4 == 0)
      return n;
    if (n % 4 == 1)
      return 1;
    if (n % 4 == 2)
      return n + 1;
    else
      return 0;
  }

  public static void main (String[] args) {

  }
}

// This code is contributed by akashish__

Python3

# Direct XOR of all numbers from 1 to n
def computeXOR(n):
    if (n % 4 is 0):
        return n
    if (n % 4 is 1):
        return 1
    if (n % 4 is 2):
        return n + 1
    else:
        return 0


# This code is contributed by akashish__

C#

using System;
public class GFG
{

  // Direct XOR of all numbers from 1 to n
  public static int computeXOR(int n)
  {

    if (n % 4 == 0)

      return n;

    if (n % 4 == 1)

      return 1;

    if (n % 4 == 2)

      return n + 1;

    else

      return 0;

  }
  public static void Main(){}


}

// This code is contributed by akashish__

Javascript

<script>

// Direct XOR of all numbers from 1 to n
function computeXOR(n)
{
    if (n % 4 == 0)
        return n;
    if (n % 4 == 1)
        return 1;
    if (n % 4 == 2)
        return n + 1;
    else
        return 0;
}

// This code is contributed by Shubham Singh

</script>

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

Refer Compute XOR from 1 to n for details.

2. Count of numbers (x) smaller than or equal to n such that n+x = n^x:

The count of such numbers x can be counted using the following mathematical trick. 

The count = pow(2, count of zero bits).

C++

// Count of numbers (x) smaller than or equal to n such that n+x = n^x:
// here unset bits means zero bits
#include <bits/stdc++.h>
using namespace std;

// function to count number of values less than
// equal to n that satisfy the given condition
int countValues(int n)
{
    // unset_bits keeps track of count of un-set
    // bits in binary representation of n
    int unset_bits=0;
    while (n)
    {
        if ((n & 1) == 0)
            unset_bits++;
        n=n>>1;
    }

    // Return 2 ^ unset_bits i.e. pow(2,count of zero bits)
    return 1 << unset_bits;
}

// Driver code
int main()
{
    int n = 15;
    cout << countValues(n);
    return 0;
}

// contributed by akashish__

Java

// Count of numbers (x) smaller than or equal to n such that n+x = n^x:
// here unset bits means zero bits
import java.io.*;
class GFG
{

  // function to count number of values less than
  // equal to n that satisfy the given condition
  public static int countValues(int n)
  {
    // unset_bits keeps track of count of un-set
    // bits in binary representation of n
    int unset_bits = 0;
    while (n > 0)
    {
      if ((n & 1) == 0)
        unset_bits++;
      n = n>>1;
    }

    // Return 2 ^ unset_bits i.e. pow(2,count of zero bits)
    return 1 << unset_bits;
  }

  // Driver code
  public static void main (String[] args) {
    int n = 15;
    System.out.print(countValues(n));
  }
}

// This code is contributed by poojaagrawal2.

Python3

# Count of numbers (x) smaller than or equal to n such that n+x = n^x:
# here unset bits means zero bits

# function to count number of values less than
# equal to n that satisfy the given condition
def countValues(n):

    # unset_bits keeps track of count of un-set
    # bits in binary representation of n
    unset_bits=0
    while (n):
        if ((n & 1) == 0):
            unset_bits+=1
        n=n>>1

    # Return 2 ^ unset_bits i.e. pow(2,count of zero bits)
    return 1 << unset_bits

# Driver code
n = 15
print(countValues(n))

# This code is contributed by akashish__

C#

// C# code to implement the approach
using System;
using System.Collections.Generic;

class GFG {

  // Count of numbers (x) smaller than or equal to n such that n+x = n^x:
  // here unset bits means zero bits

  // function to count number of values less than
  // equal to n that satisfy the given condition
  static int countValues(int n)
  {

    // unset_bits keeps track of count of un-set
    // bits in binary representation of n
    int unset_bits = 0;
    while (n > 0)
    {
      if ((n & 1) == 0)
        unset_bits++;
      n = n>>1;
    }

    // Return 2 ^ unset_bits i.e. pow(2,count of zero bits)
    return 1 << unset_bits;
  }

  // Driver code
  public static void Main()
  {
    int n = 15;
    Console.Write(countValues(n));
  }
}

// This code is contributed by agrawalpoojaa976.

Javascript

// Count of numbers (x) smaller than or equal to n such that n+x = n^x:
// here unset bits means zero bits

// function to count number of values less than
// equal to n that satisfy the given condition
function countValues(n)
{
    // unset_bits keeps track of count of un-set
    // bits in binary representation of n
    let unset_bits=0;
    while (n)
    {
        if ((n & 1) == 0)
            unset_bits++;
        n=n>>1;
    }

    // Return 2 ^ unset_bits i.e. pow(2,count of zero bits)
    return 1 << unset_bits;
}

// Driver code
    let n = 15;
    document.write(countValues(n));

Output

1

Refer Equal Sum and XOR for details.

Time complexity: O(log n)

Auxiliary Space: O(1)

3. How to know if a number is a power of 2?

This can be solved based on the following fact:

If a number N is a power of 2, then the bitwise AND of N and N-1 will be 0. But this will not work if N is 0. So just check these two conditions, if any of these two conditions is true.

Refer check if a number is power of two for details.

Below is the implementation of the above approach.

CPP

//  Function to check if x is power of 2
bool isPowerOfTwo(int x)
{
     // First x in the below expression is
     // for  the case when x is 0
     return x && (!(x & (x - 1)));
}

Java

/*package whatever //do not write package name here */
import java.io.*;

class GFG {

  //  Function to check if x is power of 2
  static public boolean isPowerOfTwo(int x)
  {

    // First x in the below expression is
    // for  the case when x is 0
    return (x != 0) && ((x & (x - 1)) == 0);
  }

  public static void main (String[] args) {


  }
}
// contributed by akashish__

Python3

#  Function to check if x is power of 2
def isPowerOfTwo(x):

  # First x in the below expression is
  # for  the case when x is 0
  return x and (not(x & (x - 1)))

# This code is contributed by akashish__

C#

using System;

public class GFG{

  //  Function to check if x is power of 2
static public bool isPowerOfTwo(int x)
{
     // First x in the below expression is
     // for  the case when x is 0
       return (x != 0) && ((x & (x - 1)) == 0);
}

    static public void Main (){

    }
}

// This code is contributed by akashish__

Javascript

//  Function to check if x is power of 2
function isPowerOfTwo(x)
{
     // First x in the below expression is
     // for  the case when x is 0
     return x && (!(x & (x - 1)));
}

// This code is contributed by akashish__

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

4. Find XOR of all subsets of a set

We can do it in O(1) time. The answer is always 0 if the given set has more than one element. For sets with a single element, the answer is the value of the single element. 

Refer XOR of the XOR’s of all subsets for details.

5. Find the number of leading, trailing zeroes and number of 1’s

We can quickly find the number of leading, trailing zeroes and number of 1’s in a binary code of an integer in C++ using GCC. 

It can be done by using inbuilt functions i.e.

Number of leading zeroes: __builtin_clz(x)
Number of trailing zeroes : __builtin_ctz(x)
Number of 1-bits: __builtin_popcount(x) 

Refer GCC inbuilt functions for details.

6. Convert binary code directly into an integer in C++

CPP

// Conversion into Binary code

#include <iostream>
using namespace std;

int main()
{
    auto number = 0b011;
    cout << number;
    return 0;
}

Java

/*package whatever //do not write package name here */
// Conversion into Binary code
import java.io.*;

class GFG {
    public static void main(String[] args)
    {
        int number = 0b011;
        System.out.println(number);
    }
}

// This code is contributed by akashish__

Python3

# Python Code
number = 0b011
print(number)

# This code is contributed by akashish__

C#

// Conversion into Binary code

using System;

public class GFG {

    static public void Main()
    {

        // Code
        int number = 0b011;
        Console.WriteLine(number);
    }
}

// This code is contributed by karthik

Javascript

// Conversion into Binary code

let number = 0b011;
console.log(number);

Output

3

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

7. The Quickest way to swap two numbers:

Two numbers can be swapped easily using the following bitwise operations:

a ^= b;
b ^= a; 
a ^= b;

C++

#include <iostream>
using namespace std;

int main()
{
    int a = 5;
      int b = 7;
      cout<<"Before Swapping, a = "<<a<<" "<<"b = "<<b<<endl;
    a ^= b;
    b ^= a;
    a ^= b;
      cout<<"After Swapping, a = "<<a<<" "<<"b = "<<b<<endl;
    return 0;
}

// This code is contributed by akashish__

Java

/*package whatever //do not write package name here */
import java.io.*;
import java.util.*;

class GFG {
  public static void main(String[] args)
  {
    int a = 5;
    int b = 7;
    System.out.print("Before Swapping, a = ");
    System.out.print(a);
    System.out.print(" ");
    System.out.print("b = ");
    System.out.print(b);
    System.out.println("");
    a ^= b;
    b ^= a;
    a ^= b;
    System.out.print("After Swapping, a = ");
    System.out.print(a);
    System.out.print(" ");
    System.out.print("b = ");
    System.out.print(b);
  }
}

// This code is contributed by akashish__

Python3

a = 5
b = 7
print("Before Swapping, a = ",a," ","b = ",b)
a ^= b
b ^= a
a ^= b
print("After Swapping, a = ",a," ","b = ",b)

# This code is contributed by akashish__

C#

using System;

public class GFG {

    static public void Main()
    {

        int a = 5;
        int b = 7;
        Console.WriteLine("Before Swapping, a = " + a + " "
                          + "b = " + b);
        a ^= b;
        b ^= a;
        a ^= b;
        Console.WriteLine("After Swapping, a = " + a + " "
                          + "b = " + b);
    }
}
// This code is contributed by akashish__

Javascript

let a = 5;
let b = 7;
console.log("Before Swapping, a = " , a , " " , "b = " , b);
a ^= b;
b ^= a;
a ^= b;
console.log("After Swapping, a = " , a , " " , "b = " , b);

// This code is contributed by akashish__

Output

Before Swapping, a = 5 b = 7
After Swapping, a = 7 b = 5

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

Refer swap two numbers for more details.  

8. Finding the most significant set bit (MSB):

We can find the most significant set bit in O(1) time for a fixed size integer. For example below code is for 32-bit integer.  

C++

int setBitNumber(int n)
{
    // Below steps set bits after
    // MSB (including MSB)

    // Suppose n is 273 (binary
    // is 100010001). It does following
    // 100010001 | 010001000 = 110011001
    n |= n >> 1;

    // This makes sure 4 bits
    // (From MSB and including MSB)
    // are set. It does following
    // 110011001 | 001100110 = 111111111
    n |= n >> 2;

    n |= n >> 4;
    n |= n >> 8;
    n |= n >> 16;

    // Increment n by 1 so that
    // there is only one set bit
    // which is just before original
    // MSB. n now becomes 1000000000
    n = n + 1;

    // Return original MSB after shifting.
    // n now becomes 100000000
    return (n >> 1);
}

Java

/*package whatever //do not write package name here */

import java.io.*;

class GFG {

  public static int setBitNumber(int n)
  {
    // Below steps set bits after
    // MSB (including MSB)

    // Suppose n is 273 (binary
    // is 100010001). It does following
    // 100010001 | 010001000 = 110011001
    n |= n >> 1;

    // This makes sure 4 bits
    // (From MSB and including MSB)
    // are set. It does following
    // 110011001 | 001100110 = 111111111
    n |= n >> 2;

    n |= n >> 4;
    n |= n >> 8;
    n |= n >> 16;

    // Increment n by 1 so that
    // there is only one set bit
    // which is just before original
    // MSB. n now becomes 1000000000
    n = n + 1;

    // Return original MSB after shifting.
    // n now becomes 100000000
    return (n >> 1);
  }


  public static void main (String[] args) {
  }
}

// This code is contributed by akashish__

Python3

def setBitNumber(n):
    # Below steps set bits after
    # MSB (including MSB)

    # Suppose n is 273 (binary
    # is 100010001). It does following
    # 100010001 | 010001000 = 110011001
    n |= n >> 1

    # This makes sure 4 bits
    # (From MSB and including MSB)
    # are set. It does following
    # 110011001 | 001100110 = 111111111
    n |= n >> 2

    n |= n >> 4
    n |= n >> 8
    n |= n >> 16

    # Increment n by 1 so that
    # there is only one set bit
    # which is just before original
    # MSB. n now becomes 1000000000
    n = n + 1

    # Return original MSB after shifting.
    # n now becomes 100000000
    return (n >> 1)

# This code is contributed by akashish__

C#

using System;

public class GFG{

  public static int setBitNumber(int n)
{
    // Below steps set bits after
    // MSB (including MSB)

    // Suppose n is 273 (binary
    // is 100010001). It does following
    // 100010001 | 010001000 = 110011001
    n |= n >> 1;

    // This makes sure 4 bits
    // (From MSB and including MSB)
    // are set. It does following
    // 110011001 | 001100110 = 111111111
    n |= n >> 2;

    n |= n >> 4;
    n |= n >> 8;
    n |= n >> 16;

    // Increment n by 1 so that
    // there is only one set bit
    // which is just before original
    // MSB. n now becomes 1000000000
    n = n + 1;

    // Return original MSB after shifting.
    // n now becomes 100000000
    return (n >> 1);
}

    static public void Main (){

        // Code
    }
}

// This code is contributed by akashish__

Javascript

function setBitNumber(n)
{
    // Below steps set bits after
    // MSB (including MSB)

    // Suppose n is 273 (binary
    // is 100010001). It does following
    // 100010001 | 010001000 = 110011001
    n |= n >> 1;

    // This makes sure 4 bits
    // (From MSB and including MSB)
    // are set. It does following
    // 110011001 | 001100110 = 111111111
    n |= n >> 2;

    n |= n >> 4;
    n |= n >> 8;
    n |= n >> 16;

    // Increment n by 1 so that
    // there is only one set bit
    // which is just before original
    // MSB. n now becomes 1000000000
    n = n + 1;

    // Return original MSB after shifting.
    // n now becomes 100000000
    return (n >> 1);
}

// This code is contributed by akashish__

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

Refer Find most significant set bit of a number for details. 

9. Check if a number has bits in an alternate pattern

We can quickly check if bits in a number are in an alternate pattern (like 101010). 

Compute bitwise XOR (XOR denoted using ^) of n and (n >> 1). If n has an alternate pattern, then n ^ (n >> 1) operation will produce a number having all bits set.

Below is the implementation of the above approach.

C++

// function to check if all the bits
// are set or not in the binary
// representation of 'n'
static bool allBitsAreSet(int n)
{
    // if true, then all bits are set
    if (((n + 1) & n) == 0)
        return true;

    // else all bits are not set
    return false;
}

// Function to check if a number
// has bits in alternate pattern
bool bitsAreInAltOrder(unsigned int n)
{
    unsigned int num = n ^ (n >> 1);

    // To check if all bits are set in 'num'
    return allBitsAreSet(num);
}

Java

/*package whatever //do not write package name here */
import java.io.*;

class GFG {

  // function to check if all the bits
  // are set or not in the binary
  // representation of 'n'
  public static boolean allBitsAreSet(long n)
  {

    // if true, then all bits are set
    if (((n + 1) & n) == 0)
      return true;

    // else all bits are not set
    return false;
  }

  // Function to check if a number
  // has bits in alternate pattern
  public static boolean bitsAreInAltOrder(long n)
  {
    long num = n ^ (n >> 1);

    // To check if all bits are set in 'num'
    return allBitsAreSet(num);
  }
  public static void main (String[] args) {

  }
}

// This code is contributed by akashish__

Python3

# function to check if all the bits
# are set or not in the binary
# representation of 'n'
def allBitsAreSet(n):
  # if true, then all bits are set
  if (((n + 1) & n) == 0):
    return True

  # else all bits are not set
  return False

# Function to check if a number
# has bits in alternate pattern
def bitsAreInAltOrder(n):
  num = n ^ (n >> 1)

  # To check if all bits are set in 'num'
  return allBitsAreSet(num)


# This code is contributed by akashish__

C#

using System;
public class GFG
{

    // function to check if all the bits
    // are set or not in the binary
    // representation of 'n'
    public static bool allBitsAreSet(uint n)
    {

        // if true, then all bits are set
        if (((n + 1) & n) == 0)
            return true;

        // else all bits are not set
        return false;
    }

    // Function to check if a number
    // has bits in alternate pattern
    public static bool bitsAreInAltOrder(uint n)
    {
        uint num = n ^ (n >> 1);

        // To check if all bits are set in 'num'
        return allBitsAreSet(num);
    }

    static public void Main() {}
}

// This code is contributed by akashish__

Javascript

// function to check if all the bits
// are set or not in the binary
// representation of 'n'
function allBitsAreSet(n)
{
    // if true, then all bits are set
    if (((n + 1) & n) == 0)
        return true;

    // else all bits are not set
    return false;
}

// Function to check if a number
// has bits in alternate pattern
function bitsAreInAltOrder(n)
{
    let num = n ^ (n >> 1);

    // To check if all bits are set in 'num'
    return allBitsAreSet(num);
}

// This code is contributed by akashish__

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

Refer check if a number has bits in alternate pattern for details.

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