All about Bit Manipulation
Last Updated :
18 Apr, 2023
Bit Manipulation is a technique used in a variety of problems to get the solution in an optimized way. This technique is very effective from a Competitive Programming point of view. It is all about Bitwise Operators which directly works upon binary numbers or bits of numbers that help the implementation fast. Below are the Bitwise Operators that are used:
- Bitwise AND (&)
- Bitwise OR (|)
- Bitwise XOR (^)
- Bitwise NOT (!)
All data in computer programs are internally stored as bits, i.e., as numbers 0 and 1.
Bit representation
In programming, an n-bit integer is internally stored as a binary number that consists of n bits. For example, the C++ type int is a 32-bit type, which means that every int number consists of 32 bits.
The int number 43 = 00000000000000000000000000101011
The bits in the representation are indexed from right to left. To convert a bit representation bk ···b2 b1 b0 into a number, we can use the formula
bk2k +...+ b222 + b121 + b020.
E.g., 1·25+1·23 +1·21 +1·20 = 43.
The bit representation of a number is either signed or unsigned.
Usually, a signed representation is used, which means that both negative and positive numbers can be represented.
A signed variable of n bits can contain any integer between -2n-1 and 2n-1 - 1
The int type in C++ is a signed type, so an int variable can contain any integer between -231 and 231 - 1.
The first bit in a signed representation is the sign of the number, 0 for non-negative numbers and 1 for negative numbers and the remaining n−1 bits contain the magnitude of the number.
Two’s complement is used, which means that the opposite number of a number is calculated by first inverting all the bits in the number, and then increasing the number by one.
The bit representation of the int number −43 is 11111111111111111111111111010101
In an unsigned representation, only non-negative numbers can be used, but the upper bound for the values is larger.
An unsigned variable of n bits can contain any integer between 0 and 2n −1.
In C++, an unsigned int variable can contain any integer between 0 and 232 −1.
There is a connection between the representations:
A signed number −x equals an unsigned number 2n − x.
For example, the following pseudo-code snippet shows that the signed number
x = −43 equals the unsigned number y = 232 −43:
the pseudo-code snippet shows that the signed number
If a number is larger than the upper bound of the bit representation, the number will overflow. In a signed representation, the next number after 2n-1 - 1 is -2n-1, and in an unsigned representation, the next number after 2n -1 is 0. For example, consider the following pseudo-code snippet:
Initially, the value of x is 231 −1. This is the largest value that can be stored in an int variable, so the next number after 231 −1 is −231 .
Learn more about Bitwise Operators in this article. Below are some common bit operations that are frequently used in programming:
Bitwise Operations:
Below is the table to illustrate the result when the operation is performed using Bitwise Operators. Here 0s or 1s mean a sequence of 0 or 1 respectively.
| Operators | Operations | Result |
|---|
| XOR | X ^ 0s | X |
|---|
| XOR | X ^ 1s | ~X |
|---|
| XOR | X ^ X | 0 |
|---|
| AND | X & 0s | 0 |
|---|
| AND | X & 1s | X |
|---|
| AND | X & X | X |
|---|
| OR | X | 0s | X |
|---|
| OR | X | 1s | 1s |
|---|
| OR | X | X | X |
|---|
Get Bit:
This method is used to find the bit at a particular position(say i) of the given number N. The idea is to find the Bitwise AND of the given number and 2i that can be represented as (1 << i). If the value return is 1 then the bit at the ith position is set. Otherwise, it is unset.
Below is the pseudo-code for the same:
C++
// Function to get the bit at the
// ith position
bool getBit(int num, int i)
{
// Return true if the bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// Function to get the bit at the
// ith position
static boolean getBit(int num, int i)
{
// Return true if the bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// This code is contributed by rishavmahato348.
# Function to get the bit at the
# ith position
def getBit(num, i):
# Return true if the bit is
# set. Otherwise return false
return ((num & (1 << i)) != 0)
# This code is contributed by shivani
// Function to get the bit at the
// ith position
static bool getBit(int num, int i)
{
// Return true if the bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// This code is contributed by subhammahato348.
<script>
// Function to get the bit at the
// ith position
function getBit(num, i)
{
// Return true if the bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// This code is contributed by Ankita saini
</script>
Set Bit:
This method is used to set the bit at a particular position(say i) of the given number N. The idea is to update the value of the given number N to the Bitwise OR of the given number N and 2i that can be represented as (1 << i). If the value return is 1 then the bit at the ith position is set. Otherwise, it is unset.
Below is the pseudo-code for the same:
C++
// Function to set the ith bit of the
// given number num
int setBit(int num, int i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// Function to set the ith bit of the
// given number num
static int setBit(int num, int i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// This code is contributed by subhammahato348
# Function to set the ith bit of the
# given number num
def setBit(num, i):
# Sets the ith bit and return
# the updated value
return num | (1 << i)
# This code is contributed by kirti
// Function to set the ith bit of the
// given number num
static int setBit(int num, int i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// Function to set the ith bit of the
// given number num
function setBit(num, i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
Clear Bit:
This method is used to clear the bit at a particular position(say i) of the given number N. The idea is to update the value of the given number N to the Bitwise AND of the given number N and the compliment of 2i that can be represented as ~(1 << i). If the value return is 1 then the bit at the ith position is set. Otherwise, it is unset.
Below is the pseudo-code for the same:
C++
// Function to clear the ith bit of
// the given number num
int clearBit(int num, int i)
{
// Create the mask for the ith
// bit unset
int mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// Function to clear the ith bit of
// the given number num
static int clearBit(int num, int i)
{
// Create the mask for the ith
// bit unset
int mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// This code is contributed by subham348
# Function to clear the ith bit of
# the given number num
def clearBit(num, i):
# Create the mask for the ith
# bit unset
mask = ~(1 << i)
# Return the updated value
return num & mask
# This code is contributed by subhammahato348
// Function to clear the ith bit of
// the given number num
static int clearBit(int num, int i)
{
// Create the mask for the ith
// bit unset
int mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// This code is contributed by Ankita Saini
// Function to clear the ith bit of
// the given number num
function clearBit(num, i)
{
// Create the mask for the ith
// bit unset
let mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// This code is contributed by souravmahato348.
Below is the program that implements the above functionalities:
C++
// C++ program to implement all the
// above functionalities
#include <bits/stdc++.h>
using namespace std;
// Function to get the bit at the
// ith position
bool getBit(int num, int i)
{
// Return true if the ith bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// Function to set the ith bit of the
// given number num
int setBit(int num, int i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// Function to clear the ith bit of
// the given number num
int clearBit(int num, int i)
{
// Create the mask for the ith
// bit unset
int mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// Driver Code
int main()
{
// Given number N
int N = 70;
cout << "The bit at the 3rd position from LSB is: "
<< (getBit(N, 3) ? '1' : '0')
<< endl;
cout << "The value of the given number "
<< "after setting the bit at "
<< "LSB is: "
<< setBit(N, 0) << endl;
cout << "The value of the given number "
<< "after clearing the bit at "
<< "LSB is: "
<< clearBit(N, 0) << endl;
return 0;
}
// Java program to implement all the
// above functionalities
import java.io.*;
class GFG{
// Function to get the bit at the
// ith position
static boolean getBit(int num, int i)
{
// Return true if the ith bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// Function to set the ith bit of the
// given number num
static int setBit(int num, int i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// Function to clear the ith bit of
// the given number num
static int clearBit(int num, int i)
{
// Create the mask for the ith
// bit unset
int mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// Driver Code
public static void main(String[] args)
{
// Given number N
int N = 70;
System.out.println("The bit at the 3rd position from LSB is: " +
(getBit(N, 3) ? '1' : '0'));
System.out.println("The value of the given number " +
"after setting the bit at " +
"LSB is: " + setBit(N, 0));
System.out.println("The value of the given number " +
"after clearing the bit at " +
"LSB is: " + clearBit(N, 0));
}
}
// This code is contributed by souravmahato348
# Python program to implement all the
# above functionalities
# Function to get the bit at the
# ith position
def getBit( num, i):
# Return true if the ith bit is
# set. Otherwise return false
return ((num & (1 << i)) != 0)
# Function to set the ith bit of the
# given number num
def setBit( num, i):
# Sets the ith bit and return
# the updated value
return num | (1 << i)
# Function to clear the ith bit of
# the given number num
def clearBit( num, i):
# Create the mask for the ith
# bit unset
mask = ~(1 << i)
# Return the updated value
return num & mask
# Driver Code
# Given number N
N = 70
print"The bit at the 3rd position from LSB is: " , 1 if (getBit(N, 3)) else '0'
print"The value of the given number" , "after setting the bit at","LSB is: " , setBit(N, 0)
print"The value of the given number" , "after clearing the bit at","LSB is: " , clearBit(N, 0)
# This code is contributed by shivanisinghss2110
// C# program to implement all the
// above functionalities
using System;
class GFG {
// Function to get the bit at the
// ith position
static bool getBit(int num, int i)
{
// Return true if the ith bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// Function to set the ith bit of the
// given number num
static int setBit(int num, int i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// Function to clear the ith bit of
// the given number num
static int clearBit(int num, int i)
{
// Create the mask for the ith
// bit unset
int mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// Driver Code
public static void Main()
{
// Given number N
int N = 70;
Console.WriteLine("The bit at the 3rd position form LSB is: "
+ (getBit(N, 3) ? '1' : '0'));
Console.WriteLine("The value of the given number "
+ "after setting the bit at "
+ "LSB is: " + setBit(N, 0));
Console.WriteLine("The value of the given number "
+ "after clearing the bit at "
+ "LSB is: " + clearBit(N, 0));
}
}
// This code is contributed by rishavmahato348.
<script>
// Javascript program to implement all
// the above functionalities
// Function to get the bit at the
// ith position
function getBit(num, i)
{
// Return true if the ith bit is
// set. Otherwise return false
return ((num & (1 << i)) != 0);
}
// Function to set the ith bit of the
// given number num
function setBit(num, i)
{
// Sets the ith bit and return
// the updated value
return num | (1 << i);
}
// Function to clear the ith bit of
// the given number num
function clearBit(num, i)
{
// Create the mask for the ith
// bit unset
let mask = ~(1 << i);
// Return the updated value
return num & mask;
}
// Driver code
// Given number N
let N = 70;
document.write("The bit at the 3rd position from LSB is: " +
(getBit(N, 3) ? '1' : '0') + "</br>");
document.write("The value of the given number " +
"after setting the bit at " +
"LSB is: " + setBit(N, 0) + "</br>");
document.write("The value of the given number " +
"after clearing the bit at " +
"LSB is: " + clearBit(N, 0) + "</br>");
// This code is contributed by divyeshrabadiya07
</script>
OutputThe bit at the 3rd position from LSB is: 0
The value of the given number after setting the bit at LSB is: 71
The value of the given number after clearing the bit at LSB is: 70
Time Complexity: O(1)
Auxiliary Space: O(1)
Application of Bitwise Operator
- Bitwise operations are prominent in embedded systems, control systems, etc where memory(data transmission/data points) is still an issue.
- They are also useful in networking where it is important to reduce the amount of data, so booleans are packed together. Packing them together and taking them apart use bitwise operations and shift instructions.
- Bitwise operations are also heavily used in the compression and encryption of data.
- Useful in graphics programming, older GUIs are heavily dependent on bitwise operations like XOR(^) for selection highlighting and other overlays.
Similar Reads
Bitwise Algorithms
Bitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Introduction to Bitwise Algorithms - Data Structures and Algorithms Tutorial
Bit stands for binary digit. A bit is the basic unit of information and can only have one of two possible values that is 0 or 1. In our world, we usually with numbers using the decimal base. In other words. we use the digit 0 to 9 However, there are other number representations that can be quite use
15+ min read
Bitwise Operators in C
In C, bitwise operators are used to perform operations directly on the binary representations of numbers. These operators work by manipulating individual bits (0s and 1s) in a number.The following 6 operators are bitwise operators (also known as bit operators as they work at the bit-level). They are
6 min read
Bitwise Operators in Java
In Java, Operators are special symbols that perform specific operations on one or more than one operands. They build the foundation for any type of calculation or logic in programming.There are so many operators in Java, among all, bitwise operators are used to perform operations at the bit level. T
6 min read
Python Bitwise Operators
Python bitwise operators are used to perform bitwise calculations on integers. The integers are first converted into binary and then operations are performed on each bit or corresponding pair of bits, hence the name bitwise operators. The result is then returned in decimal format.Note: Python bitwis
5 min read
JavaScript Bitwise Operators
In JavaScript, a number is stored as a 64-bit floating-point number but bitwise operations are performed on a 32-bit binary number. To perform a bit-operation, JavaScript converts the number into a 32-bit binary number (signed) and performs the operation and converts back the result to a 64-bit numb
5 min read
All about Bit Manipulation
Bit Manipulation is a technique used in a variety of problems to get the solution in an optimized way. This technique is very effective from a Competitive Programming point of view. It is all about Bitwise Operators which directly works upon binary numbers or bits of numbers that help the implementa
14 min read
What is Endianness? Big-Endian & Little-Endian
Computers operate using binary code, a language made up of 0s and 1s. This binary code forms the foundation of all computer operations, enabling everything from rendering videos to processing complex algorithms. A single bit is a 0 or a 1, and eight bits make up a byte. While some data, such as cert
5 min read
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^xHow to know if a number is a power of 2?Find XOR of all
15+ min read
Easy Problems on Bit Manipulations and Bitwise Algorithms
Binary representation of a given number
Given an integer n, the task is to print the binary representation of the number. Note: The given number will be maximum of 32 bits, so append 0's to the left if the result string is smaller than 30 length.Examples: Input: n = 2Output: 00000000000000000000000000000010Input: n = 0Output: 000000000000
6 min read
Count set bits in an integer
Write an efficient program to count the number of 1s in the binary representation of an integer.Examples : Input : n = 6Output : 2Binary representation of 6 is 110 and has 2 set bitsInput : n = 13Output : 3Binary representation of 13 is 1101 and has 3 set bits[Naive Approach] - One by One CountingTh
15+ min read
Add two bit strings
Given two binary strings s1 and s2 consisting of only 0s and 1s. Find the resultant string after adding the two Binary Strings.Note: The input strings may contain leading zeros but the output string should not have any leading zeros.Examples:Input: s1 = "1101", s2 = "111"Output: 10100Explanation: "1
1 min read
Turn off the rightmost set bit
Given an integer n, turn remove turn off the rightmost set bit in it. Input: 12Output: 8Explanation : Binary representation of 12 is 00...01100. If we turn of the rightmost set bit, we get 00...01000 which is binary representation of 8Input: 7 Output: 6 Explanation : Binary representation for 7 is 0
7 min read
Rotate bits of a number
Given a 32-bit integer n and an integer d, rotate the binary representation of n by d positions in both left and right directions. After each rotation, convert the result back to its decimal representation and return both values in an array as [left rotation, right rotation].Note: A rotation (or cir
7 min read
Compute modulus division by a power-of-2-number
Given two numbers n and d where d is a power of 2 number, the task is to perform n modulo d without the division and modulo operators.Input: 6 4Output: 2 Explanation: As 6%4 = 2Input: 12 8Output: 4Explanation: As 12%8 = 4Input: 10 2Output: 0Explanation: As 10%2 = 0Approach:The idea is to leverage bi
3 min read
Find the Number Occurring Odd Number of Times
Given an array of positive integers. All numbers occur an even number of times except one number which occurs an odd number of times. Find the number in O(n) time & constant space. Examples : Input : arr = {1, 2, 3, 2, 3, 1, 3}Output : 3 Input : arr = {5, 7, 2, 7, 5, 2, 5}Output : 5 Recommended
12 min read
Program to find whether a given number is power of 2
Given a positive integer n, the task is to find if it is a power of 2 or not.Examples: Input : n = 16Output : YesExplanation: 24 = 16Input : n = 42Output : NoExplanation: 42 is not a power of 2Input : n = 1Output : YesExplanation: 20 = 1Approach 1: Using Log - O(1) time and O(1) spaceThe idea is to
12 min read
Find position of the only set bit
Given a number n containing only 1 set bit in its binary representation, the task is to find the position of the only set bit. If there are 0 or more than 1 set bits, then return -1. Note: Position of set bit '1' should be counted starting with 1 from the LSB side in the binary representation of the
8 min read
Check for Integer Overflow
Given two integers a and b. The task is to design a function that adds two integers and detects overflow during the addition. If the sum does not cause an overflow, return their sum. Otherwise, return -1 to indicate an overflow.Note: You cannot use type casting to a larger data type to check for ove
7 min read
Find XOR of two number without using XOR operator
Given two integers, the task is to find XOR of them without using the XOR operator.Examples : Input: x = 1, y = 2Output: 3Input: x = 3, y = 5Output: 6Approach - Checking each bit - O(log n) time and O(1) spaceA Simple Solution is to traverse all bits one by one. For every pair of bits, check if both
8 min read
Check if two numbers are equal without using arithmetic and comparison operators
Given two numbers, the task is to check if two numbers are equal without using Arithmetic and Comparison Operators or String functions. Method 1 : The idea is to use XOR operator. XOR of two numbers is 0 if the numbers are the same, otherwise non-zero. C++ // C++ program to check if two numbers // a
8 min read
Detect if two integers have opposite signs
Given two integers a and b, the task is to determine whether they have opposite signs. Return true if the signs of the two numbers are different and false otherwise.Examples:Input: a = -5, b = 10Output: trueExplanation: One number is negative and the other is positive, so their signs are different.I
9 min read
Swap Two Numbers Without Using Third Variable
Given two variables a and y, swap two variables without using a third variable. Examples: Input: a = 2, b = 3Output: a = 3, b = 2Input: a = 20, b = 0Output: a = 0, b = 20Input: a = 10, b = 10Output: a = 10, b = 10Table of ContentUsing Arithmetic OperatorsUsing Bitwise XORBuilt-in SwapUsing Arithmeti
6 min read
Russian Peasant (Multiply two numbers using bitwise operators)
Given two integers a and b, the task is to multiply them without using the multiplication operator. Instead of that, use the Russian Peasant Algorithm.Examples:Input: a = 2, b = 5Output: 10Explanation: Product of 2 and 5 is 10.Input: a = 6, b = 9Output: 54Explanation: Product of 6 and 9 is 54.Input:
4 min read