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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 useful such as the binary number systems.

Introduction to Bitwise Algorithms – Data Structures and Algorithms Tutorial

Unlike humans, computers have no concepts of words and numbers. They receive data encoded at the lowest level as a series of zeros and ones (0 and 1). These are called bits, and they are the basis for all the commands they receive. We’ll begin by learning about bits and then explore a few algorithms for manipulating bits. We’ll then explore a few algorithms for manipulating bits. The tutorial is meant to be an introduction to bit algorithms for programmers.

Need of the  Bitwise Algorithms :-

Bitwise algorithms are an important tool in computer science and programming because they allow for efficient manipulation and processing of binary data. Here are a few reasons why bitwise algorithms are necessary:

Space Efficiency: Bitwise algorithms can be used to represent data in a more compact form, saving space in memory and on disk.

Time Efficiency: Bitwise operations are often faster than their equivalent operations using other data types, such as integers. This can lead to significant performance gains in time-sensitive applications.

Masking and Bitwise Encoding: Bitwise algorithms can be used to mask bits of data, which is useful for data encryption and compression. Bitwise encoding is also used to encode and decode data in a compact form, which is useful for transmitting data over networks and storing data on disk.

Optimization: Bitwise algorithms can be used to optimize algorithms by allowing for efficient manipulations of data at the binary level.

Hardware Interaction: Bitwise algorithms are often used to interact directly with hardware components, such as memory, networks, and processors, as well as to manipulate binary data in machine-level programming languages, such as Assembly.

Overall, bitwise algorithms play a critical role in modern computing and are a valuable tool for efficient data manipulation and processing.

Speed: Bitwise operations are fast and efficient as they are performed directly on the binary representation of data, without the need for conversion to other data types.

Space Efficiency: Bitwise algorithms can be used to represent data in a more compact form, saving space in memory and on disk.

Masking and Bitwise Encoding: Bitwise algorithms can be used to mask bits of data, which is useful for data encryption and compression. Bitwise encoding is also used to encode and decode data in a compact form, which is useful for transmitting data over networks and storing data on disk.

Optimization: Bitwise algorithms can be used to optimize algorithms by allowing for efficient manipulations of data at the binary level.

Hardware Interaction: Bitwise algorithms are often used to interact directly with hardware components, such as memory, networks, and processors, as well as to manipulate binary data in machine-level programming languages, such as Assembly.

Complexity: Bitwise algorithms can be difficult to understand and implement, particularly for those who are not familiar with binary representations of data and bitwise operations.

Portability: Bitwise algorithms can be platform-specific and may not be portable across different systems, particularly when dealing with binary data at the machine level.

Maintenance: Bitwise algorithms can be difficult to maintain and debug, particularly when dealing with complex bitwise operations and manipulations.

Performance: While bitwise operations can be fast and efficient, they may not be the best choice for every situation. In some cases, other data types and algorithms may be more suitable for a particular task.

Basics of Bit manipulation (Bitwise Operators)

An algorithmic operation known as bit manipulation involves the manipulation of bits at the bit level (bitwise). Bit manipulation is all about these bitwise operations. They improve the efficiency of programs by being primitive, fast actions.

The computer uses this bit manipulation to perform operations like addition, subtraction, multiplication, and division are all done at the bit level. This operation is performed in the arithmetic logic unit (ALU) which is a part of a computer’s CPU. Inside the ALU, all such mathematical operations are performed.

There are different bitwise operations used in bit manipulation. These bit operations operate on the individual bits of the bit patterns. Bit operations are fast and can be used in optimizing time complexity. Some common bit operators are:

Bitwise Operator Truth Table

1. Bitwise AND Operator (&)

The bitwise AND operator is denoted using a single ampersand symbol, i.e. &. The & operator takes two equal-length bit patterns as parameters. The two-bit integers are compared. If the bits in the compared positions of the bit patterns are 1, then the resulting bit is 1. If not, it is 0.

Truth table of AND operator

Example:

Take two bit values X and Y, where X = 7= (111)2 and Y = 4 = (100)2 . Take Bitwise and of both X & y

Bitwise ANDof (7 & 4)

Implementation of AND operator:

C++

 #include using namespace std;    int main() {        int a = 7, b = 4;     int result = a & b;     cout << result << endl;        return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main (String[] args) {         int a = 7, b = 4;           int result = a & b;           System.out.println(result);     } }    // This code is contributed by lokeshmvs21.

C#

 using System;    public class GFG{        static public void Main (){       int a = 7, b = 4;       int result = a & b;       Console.WriteLine(result);     } }    // This code is contributed by akashish__

Python3

 a = 7 b = 4 result = a & b print(result) # This code is contributed by akashish__

Javascript

 let a = 7, b = 4; let result = a & b; console.log(result); // This code is contributed by akashish__

Output

4

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

2 â€‹Bitwise OR Operator (|)

The | Operator takes two equivalent length bit designs as boundaries; if the two bits in the looked-at position are 0, the next bit is zero. If not, it is 1.

Truth table of OR operator

Example:

Take two bit values X and Y, where X = 7= (111)2 and Y = 4 = (100)2 . Take Bitwise OR of both X, y

Bitwise OR of (7 | 4)

Explanation: On the basis of truth table of bitwise OR operator we can conclude that the result of

1 | 1  = 1
1 | 0 = 1
0 | 1 = 1
0 | 0 = 0

We used the similar concept of bitwise operator that are show in the image.

Implementation of OR operator:

C++

 #include using namespace std;    int main() {        int a = 12, b = 25;     int result = a | b;     cout << result;        return 0; }

Java

 import java.io.*;    class GFG {     public static void main(String[] args)     {         int a = 12, b = 25;         int result = a | b;         System.out.println(result);     } }

Python3

 a = 12 b = 25 result = a | b print(result)    # This code is contributed by garg28harsh.

C#

 using System;    public class GFG{        static public void Main (){         int a = 12, b = 25;         int result = a | b;         Console.WriteLine(result);     } } // This code is contributed by akashish__

Javascript

 let a = 12, b = 25;     let result = a | b;     document.write(result);          // This code is contributed by garg28harsh.

Output

29

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

3. â€‹Bitwise XOR Operator (^)

The ^ operator (also known as the XOR operator) stands for Exclusive Or. Here, if bits in the compared position do not match their resulting bit is 1. i.e, The result of the bitwise XOR operator is 1 if the corresponding bits of two operands are opposite, otherwise 0.

Truth Table of Bitwise Operator XOR

Example:

Take two bit values X and Y, where X = 7= (111)2 and Y = 4 = (100)2 . Take Bitwise and of both X & y

Bitwise OR of (7 ^ 4)

Explanation: On the basis of truth table of bitwise XOR operator we can conclude that the result of

1 ^ 1  = 0
1 ^ 0 = 1
0 ^ 1 = 1
0 ^ 0 = 0

We used the similar concept of bitwise operator that are show in the image.

Implementation of XOR operator:

C++

 #include using namespace std;    int main() {        int a = 12, b = 25;     cout << (a ^ b);     return 0; }

Java

 import java.io.*;    class GFG {     public static void main(String[] args)     {         int a = 12, b = 25;         int result = a ^ b;         System.out.println(result);     } }    // This code is contributed by garg28harsh.

Python3

 a = 12 b = 25 result = a ^ b print(result)    # This code is contributed by garg28harsh.

C#

 // C# Code    using System;    public class GFG {        static public void Main()     {            // Code         int a = 12, b = 25;         int result = a ^ b;         Console.WriteLine(result);     } }    // This code is contributed by lokesh

Javascript

 let a = 12; let b = 25; console.log((a ^ b)); // This code is contributed by akashish__

Output

21

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

4. â€‹Bitwise NOT Operator (!~)

All the above three bitwise operators are binary operators (i.e, requiring two operands in order to operate). Unlike other bitwise operators, this one requires only one operand to operate.

The bitwise Not Operator takes a single value and returns its oneâ€™s complement. The oneâ€™s complement of a binary number is obtained by toggling all bits in it, i.e, transforming the 0 bit to 1 and the 1 bit to 0.

Truth Table of Bitwise Operator NOT

Example:

Take two bit values X and Y, where X = 5= (101)2 . Take Bitwise NOT of X.

Explanation: On the basis of truth table of bitwise NOT operator we can conclude that the result of

~1  = 0
~0 = 1

We used the similar concept of bitwise operator that are show in the image.

Implementation of NOT operator:

C++

 #include using namespace std;    int main() {        int a = 0;     cout << "Value of a without using NOT operator: " << a;     cout << "\nInverting using NOT operator (with sign bit): " << (~a);     cout << "\nInverting using NOT operator (without sign bit): " << (!a);        return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {   public static void main(String[] args)   {     int a = 0;     System.out.println(       "Value of a without using NOT operator: " + a);     System.out.println(       "Inverting using NOT operator (with sign bit): "       + (~a));     if (a != 1)       System.out.println(       "Inverting using NOT operator (without sign bit): 1");     else       System.out.println(       "Inverting using NOT operator (without sign bit): 0");   } }    // This code is contributed by lokesh.

Python3

 a = 0 print("Value of a without using NOT operator: " , a) print("Inverting using NOT operator (with sign bit): " , (~a)) print("Inverting using NOT operator (without sign bit): " , int(not(a))) #  This code is contributed by akashish__

C#

 using System;    public class GFG {      static public void Main()   {        int a = 0;     Console.WriteLine(       "Value of a without using NOT operator: " + a);     Console.WriteLine(       "Inverting using NOT operator (with sign bit): "       + (~a));     if (a != 1)       Console.WriteLine(       "Inverting using NOT operator (without sign bit): 1");     else       Console.WriteLine(       "Inverting using NOT operator (without sign bit): 0");   } }    // This code is contributed by akashish__

Javascript

 let a =0; document.write("Value of a without using NOT operator: " + a); document.write( "Inverting using NOT operator (with sign bit): " + (~a)); if(!a) document.write( "Inverting using NOT operator (without sign bit): 1" ); else document.write( "Inverting using NOT operator (without sign bit): 0" );

Output

Value of a without using NOT operator: 0
Inverting using NOT operator (with sign bit): -1
Inverting using NOT operator (without sign bit): 1

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

5. Left-Shift (<<)

The left shift operator is denoted by the double left arrow key (<<). The general syntax for left shift is shift-expression << k. The left-shift operator causes the bits in shift expression to be shifted to the left by the number of positions specified by k. The bit positions that the shift operation has vacated are zero-filled.

Note: Every time we shift a number towards the left by 1 bit it multiply that number by 2.

Logical left Shift

Example:

Input: Left shift of 5 by 1.
Binary representation of 5 = 00101 and Left shift of 001012 by 1 (i.e, 00101 << 1)

Left shift of 5 by 1

Output: 10
Explanation: All bit of 5 will be shifted by 1 to left side and this result in 010102, Which is equivalent to 10

Input: Left shift of 5 by 2.
Binary representation of 5 = 00101 and Left shift of 001012 by 1 (i.e, 00101 << 2)

Left shift of 5 by 2

Output: 20
Explanation: All bit of 5 will be shifted by 1 to left side and this result in 101002, Which is equivalent to 20

Input: Left shift of 5 by 3.
Binary representation of 5 = 00101 and Left shift of 001012 by 1 (i.e, 00101 << 3)

Left shift of 5 by 3

Output: 40
Explanation: All bit of 5 will be shifted by 1 to left side and this result in 010002, Which is equivalent to 40

Implementation of Left shift operator:

C++

 #include using namespace std;    int main() {     unsigned int num1 = 1024;        bitset<32> bt1(num1);     cout << bt1 << endl;        unsigned int num2 = num1 << 1;     bitset<32> bt2(num2);     cout << bt2 << endl;        unsigned int num3 = num1 << 2;     bitset<16> bitset13{ num3 };     cout << bitset13 << endl; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {   public static void main(String[] args)   {     int num1 = 1024;        String bt1 = Integer.toBinaryString(num1);     bt1 = String.format("%32s", bt1).replace(' ', '0');     System.out.println(bt1);        int num2 = num1 << 1;     String bt2 = Integer.toBinaryString(num2);     bt2 = String.format("%32s", bt2).replace(' ', '0');     System.out.println(bt2);        int num3 = num1 << 2;     String bitset13 = Integer.toBinaryString(num3);     bitset13 = String.format("%16s", bitset13)       .replace(' ', '0');     System.out.println(bitset13);   } }    // This code is contributed by akashish__

Python3

 # Python code for the above approach    num1 = 1024    bt1 = bin(num1)[2:].zfill(32) print(bt1)    num2 = num1 << 1 bt2 = bin(num2)[2:].zfill(32) print(bt2)    num3 = num1 << 2 bitset13 = bin(num3)[2:].zfill(16) print(bitset13)    # This code is contributed by Prince Kumar

C#

 using System;    class GFG {   public static void Main(string[] args)   {     int num1 = 1024;        string bt1 = Convert.ToString(num1, 2);     bt1 = bt1.PadLeft(32, '0');     Console.WriteLine(bt1);        int num2 = num1 << 1;     string bt2 = Convert.ToString(num2, 2);     bt2 = bt2.PadLeft(32, '0');     Console.WriteLine(bt2);        int num3 = num1 << 2;     string bitset13 = Convert.ToString(num3, 2);     bitset13 = bitset13.PadLeft(16, '0');     Console.WriteLine(bitset13);   } }    // This code is contributed by akashish__

Javascript

 // JavaScript code for the above approach    let num1 = 1024;    let bt1 = num1.toString(2).padStart(32, '0'); console.log(bt1);    let num2 = num1 << 1; let bt2 = num2.toString(2).padStart(32, '0'); console.log(bt2);    let num3 = num1 << 2; let bitset13 = num3.toString(2).padStart(16, '0'); console.log(bitset13);

Output

00000000000000000000010000000000
00000000000000000000100000000000
0001000000000000

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

6. Right-Shift (>>)

The right shift operator is denoted by the double right arrow key (>>). The general syntax for the right shift is “shift-expression >> k”. The right-shift operator causes the bits in shift expression to be shifted to the right by the number of positions specified by k. For unsigned numbers, the bit positions that the shift operation has vacated are zero-filled. For signed numbers, the sign bit is used to fill the vacated bit positions. In other words, if the number is positive, 0 is used, and if the number is negative, 1 is used.

Note: Every time we shift a number towards the right by 1 bit it divides that number by 2.

Logical Right Shift

Example:

Input: Left shift of 5 by 1.
Binary representation of 5 = 00101 and Left shift of 001012 by 1 (i.e, 00101 << 1)

Right shift of 5 by 1

Output: 10
Explanation: All bit of 5 will be shifted by 1 to left side and this result in 010102, Which is equivalent to 10

Input: Left shift of 5 by 2.
Binary representation of 5 = 00101 and Left shift of 001012 by 1 (i.e, 00101 << 2)

Right shift of 5 by 2

Output: 20
Explanation: All bit of 5 will be shifted by 1 to left side and this result in 101002, Which is equivalent to 20

Input: Left shift of 5 by 3.
Binary representation of 5 = 00101 and Left shift of 001012 by 1 (i.e, 00101 << 3)

Right shift of 5 by 3

Output: 40
Explanation: All bit of 5 will be shifted by 1 to left side and this result in 010002, Which is equivalent to 40

Implementation of Right shift operator:

C++

 #include #include    using namespace std;    int main() {     unsigned int num1 = 1024;        bitset<32> bt1(num1);     cout << bt1 << endl;        unsigned int num2 = num1 >> 1;     bitset<32> bt2(num2);     cout << bt2 << endl;        unsigned int num3 = num1 >> 2;     bitset<16> bitset13{ num3 };     cout << bitset13 << endl; }

Javascript

 // JavaScript code for the above approach    let num1 = 1024;    let bt1 = num1.toString(2).padStart(32, '0'); console.log(bt1);    let num2 = num1 >> 1; let bt2 = num2.toString(2).padStart(32, '0'); console.log(bt2);    let num3 = num1 >> 2; let bitset13 = num3.toString(2).padStart(16, '0'); console.log(bitset13); // akashish__

Output

00000000000000000000010000000000
00000000000000000000001000000000
0000000100000000

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

Application of BIT Operators

• Bit operations are used for the optimization of embedded systems.
• The Exclusive-or operator can be used to confirm the integrity of a file, making sure it has not been corrupted, especially after it has been in transit.
• Bitwise operations are used in Data encryption and compression.
• Bits are used in the area of networking, framing the packets of numerous bits which are sent to another system generally through any type of serial interface.
• Digital Image Processors use bitwise operations to enhance image pixels and to extract different sections of a microscopic image.

Important Practice Problems on Bitwise Algorithm:

1. How to Set a bit in the number?

If we want to set a bit at nth position in the number ‘num’, it can be done using the ‘OR’ operator( | ).

• First, we left shift 1 to n position via (1<<n)
• Then, use the “OR” operator to set the bit at that position. “OR” operator is used because it will set the bit even if the bit is unset previously in the binary representation of the number ‘num’.

Note: If the bit would be already set then it would remain unchanged.

Below is the implementation:

C++

 #include using namespace std; // num is the number and pos is the position // at which we want to set the bit. void set(int& num, int pos) {     // First step is shift '1', second     // step is bitwise OR     num |= (1 << pos); } int main() {     int num = 4, pos = 1;     set(num, pos);     cout << (int)(num) << endl;     return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main(String[] args)     {         int num = 4, pos = 1;         num = set(num, pos);         System.out.println(num);     }     public static int set(int num, int pos)     {         // First step is shift '1', second         // step is bitwise OR         num |= (1 << pos);         return num;     } }    // This code is contributed by geeky01adash.

Python3

 # num = number, pos = position at which we want to set the bit def set(num, pos):     # First step = Shift '1'     # Second step = Bitwise OR     num |= (1 << pos)     print(num)       num, pos = 4, 1    set(num, pos)    # This code is contributed by sarajadhav12052009

C#

 using System;    public class GFG {        static public void Main()     {         int num = 4, pos = 1;         set(num, pos);     }        // num = number, pos = position at which we want to set     // the bit     static public void set(int num, int pos)     {         // First Step: Shift '1'         // Second Step: Bitwise OR         num |= (1 << pos);         Console.WriteLine(num);     } }    // This code is contributed by sarajadhav12052009

Javascript



Output

6

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

2. How to unset/clear a bit at n’th position in the number

Suppose we want to unset a bit at nth position in number ‘num’ then we have to do this with the help of “AND” (&) operator.

• First, we left shift ‘1’ to n position via (1<<n) then we use bitwise NOT operator ‘~’ to unset this shifted ‘1’.
• Now after clearing this left shifted ‘1’ i.e making it to ‘0’ we will ‘AND'(&) with the number ‘num’ that will unset bit at nth position.

Below is the implementation:

C++

 #include using namespace std; // First step is to get a number that Â has all 1's except // the given position. void unset(int& num, int pos) {     // Second step is to bitwise and this Â number with given     // number     num &= (~(1 << pos)); } int main() {     int num = 7;     int pos = 1;     unset(num, pos);     cout << num << endl;     return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main(String[] args)     {         int num = 7, pos = 1;         num = unset(num, pos);         System.out.println(num);     }     public static int unset(int num, int pos)     {         // Second step is to bitwise and this Â number with         // given number         num = num & (~(1 << pos));         return num;     } }

Python3

 # First Step: Getting which have all '1's except the # given position       def unset(num, pos):     # Second Step: Bitwise AND this number with the given number     num &= (~(1 << pos))     print(num)       num, pos = 7, 1    unset(num, pos)

C#

 using System;    public class GFG {        static public void Main()     {         // First Step: Getting a number which have all '1's         // except the given position         int num = 7, pos = 1;         unset(num, pos);     }     static public void unset(int num, int pos)     {         // Second Step: Bitwise AND this number with the         // given number         num &= (~(1 << pos));         Console.WriteLine(num);     } }

Javascript

 // First step is to get a number that Â has all 1's except // the given position. function unset(num, pos) {     // Second step is to bitwise and this Â number with given     // number     return num &= (~(1 << pos)); }    let num = 7; let pos = 1; console.log(unset(num, pos));    // contributed by akashish__

Output

5

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

3. Toggling a bit at nth position

Toggling means to turn bit ‘on'(1) if it was ‘off'(0) and to turn ‘off'(0) if it was ‘on'(1) previously. We will be using the ‘XOR’ operator here which is this ‘^’. The reason behind the ‘XOR’ operator is because of its properties.

• Properties of ‘XOR’ operator.
• 1^1 = 0
• 0^0 = 0
• 1^0 = 1
• 0^1 = 1
• If two bits are different then the ‘XOR’ operator returns a set bit(1) else it returns an unset bit(0).

Below is the implementation:

C++

 #include using namespace std; // First step is to shift 1,Second step is to XOR with given // number void toggle(int& num, int pos) { num ^= (1 << pos); } int main() {     int num = 4;     int pos = 1;     toggle(num, pos);     cout << num << endl;     return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main(String[] args)     {         int num = 4, pos = 1;         num = toggle(num, pos);         System.out.println(num);     }     public static int toggle(int num, int pos)     {         // First step is to shift 1,Second step is to XOR         // with given number         num ^= (1 << pos);         return num;     } }    // This code is contributed by geeky01adash.

Python3

 def toggle(num, pos):     # First Step: Shifts '1'     # Second Step: XOR num     num ^= (1 << pos)     print(num)       num, pos = 4, 1    toggle(num, pos)    # This code is contributed by sarajadhav12052009

C#

 using System;    public class GFG {        static public void Main()     {         int num = 4, pos = 1;         toggle(num, pos);     }     static public void toggle(int num, int pos)     {         // First Step: Shift '1'         // Second Step: XOR num         num ^= (1 << pos);         Console.WriteLine(num);     } }    // This code is contributed by sarajadhav12052009

Javascript

 // First step is to shift 1,Second step is to XOR with given // number function toggle(num, pos){     // First Step: Shifts '1'     // Second Step: XOR num     num ^= (1 << pos)     console.log(num) }    let num = 4; let pos = 1; toggle(num, pos); // contributed by akashish__

Output

6

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

4. Checking if the bit at nth position is Set or Unset

We used the left shift (<<) operation on 1 to shift the bits to nth position and then use the & operation with number given number, and check if it is not-equals to 0.

Below is the implementation:

C++

 #include using namespace std;    bool at_position(int num, int pos) {     bool bit = num & (1 << pos);     return bit; }    int main() {     int num = 5;     int pos = 2;     bool bit = at_position(num, pos);     cout << bit << endl;     return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main(String[] args)     {         int num = 5;         int pos = 0;         int bit = at_position(num, pos);         System.out.println(bit);     }     public static int at_position(int num, int pos)     {         int bit = num & (1 << pos);         return bit;     } }

Python3

 # code def at_position(num, pos):     bit = num & (1 << pos)     return bit       num = 5 pos = 0 bit = at_position(num, pos) print(bit)

C#

 using System;    public class GFG {      public static bool at_position(int num, int pos)   {     int bit = num & (1 << pos);     if (bit == 0)       return false;     return true;   }      static public void Main()   {     int num = 5;     int pos = 2;     bool bit = at_position(num, pos);     Console.WriteLine(bit);   } }    // This code is contributed by akashish__

Javascript



Output

1

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

5. Multiply a number by 2 using the left shift operator

Below is the implementation:

C++

 #include using namespace std; int main() {     int num = 12;     int ans = num << 1;     cout << ans << endl;     return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main(String[] args)     {         int num = 12;         int ans = num << 1;         System.out.println(ans);     } }    // This code is contributed by geeky01adash.

C#

 using System;    public class GFG {        static public void Main()     {         int num = 12;         Console.WriteLine(num << 1);     } }    // This code is contributed by sarajadhav12052009

Python3

 # Python program for the above approach    num = 12 ans = num << 1 print(ans)    # This code is contributed by Shubham Singh

Javascript



Output

24

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

6. Divide a number 2 using the right shift operator

Below is the implementation:

C++

 #include using namespace std; int main() {     int num = 12;     int ans = num >> 1;     cout << ans << endl;     return 0; }

Java

 /*package whatever //do not write package name here */    import java.io.*;    class GFG {     public static void main(String[] args)     {         int num = 12;         int ans = num >> 1;         System.out.println(ans);     } }    // This code is contributed by geeky01adash.

C#

 using System;    public class GFG {        static public void Main()     {         int num = 12;         Console.WriteLine(num >> 1);     } }    // This code is contributed by sarajadhav12052009

Python3

 # Python program for the above approach    num = 12 ans = num >> 1 print(ans)    # This code is contributed by Shubham Singh

Javascript



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

7. 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.

C++

 // 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__

Python

 # num = number, pos = position at which we want to set the bit def set(num, pos):        # First step = Shift '1'     # Second step = Bitwise OR num |= (1 << pos) print(num)    num, pos = 4, 1    set(num, pos)    # This code is contributed by sarajadhav12052009

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



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

8. 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.

Below is the implementation of the above approach.

C++

 // 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

 // Function to check if x is power of 2 public static 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); }

Python

 # 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))); } // contributed by akashish__

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

9. Count Set bits in an integer

Counting set bits means, counting total number of 1â€™s in the binary representation of an integer. For this problem we go through all the bits of given number and check whether it is set or not by performing AND operation (with 1).

Below is the implementation:

C++

 // Function to calculate the number of set bits. int countBits(int n) {     // Initialising a variable count to 0.     int count = 0;     while (n) {         // If the last bit is 1, count will be incremented         // by 1 in this step.         count += n & 1;            // Using the right shift operator.         // The bits will be shifted one position to the         // right.         n >>= 1;     }     return count; }

Java

 // Function to calculate the number of set bits. public static int countBits(int n) {     // Initialising a variable count to 0.     int count = 0;     while (n > 0) {         // If the last bit is 1, count will be incremented         // by 1 in this step.         count += n & 1;            // Using the right shift operator.         // The bits will be shifted one position to the         // right.         n >>= 1;     }     return count; }

Python3

 def countBits(n):     # Initializing a variable count to 0     count = 0     while n:         # If the last bit is 1, count will be incremented by 1 in this step.         count += n & 1         # Using the right shift operator. The bits will be shifted one position to the right.         n >>= 1     return count

C#

 using System;    public class GFG {      // Function to calculate the number of set bits.   public static int countBits(int n)   {        // Initialising a variable count to 0.     int count = 0;     while (n > 0)     {          // If the last bit is 1, count will be       // incremented by 1 in this step.       count += n & 1;          // Using the right shift operator.       // The bits will be shifted one position to the       // right.       n >>= 1;     }     return count;   }      static public void Main() {} }    // This code is contributed by akashish__

Javascript

 // Function to calculate the number of set bits. function countBits(n) {     // Initialising a variable count to 0.     let count = 0;     while (n) {         // If the last bit is 1, count will be incremented         // by 1 in this step.         count += n & 1;            // Using the right shift operator.         // The bits will be shifted one position to the         // right.         n >>= 1;     }     return count; }

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

10. Position of rightmost set bit

The idea is to unset the rightmost bit of number n and XOR the result with n. Then the rightmost set bit in n will be the position of the only set bit in the result. Note that if n is odd, we can directly return 1 as the first bit is always set for odd numbers.

Example:
The number 20 in binary is 00010100, and the position of the rightmost set bit is 3.

00010100    &               (n = 20)
00010011                     (n-1 = 19)
——————-
00010000    ^                (XOR result number with n)
00010100
——————-
00000100 ——->  rightmost set bit will tell us the position

Below is the implementation:

C++

 // Returns the position of the rightmost set bit of `n` int positionOfRightmostSetBit(int n) {     // if the number is odd, return 1     if (n & 1) {         return 1;     }        // unset rightmost bit and xor with the number itself     n = n ^ (n & (n - 1));        // find the position of the only set bit in the result;     // we can directly return `log2(n) + 1` from the     // function     int pos = 0;     while (n) {         n = n >> 1;         pos++;     }     return pos; }

Java

 // Returns the position of the rightmost set bit of `n` public static int positionOfRightmostSetBit(int n) {     // if the number is odd, return 1     if ((n & 1) != 0) {         return 1;     }        // unset rightmost bit and xor with the number itself     n = n ^ (n & (n - 1));        // find the position of the only set bit in the result;     // we can directly return `log2(n) + 1` from the     // function     int pos = 0;     while (n != 0) {         n = n >> 1;         pos++;     }        return pos; }

Python

 # Returns the position of the rightmost set bit of `n`       def positionOfRightmostSetBit(n):   # if the number is odd, return 1     if n & 1:         return 1        # unset rightmost bit and xor with the number itself     n = n ^ (n & (n - 1))        # find the position of the only set bit in the result;     # we can directly return `log2(n) + 1` from the function     pos = 0     while n:         n = n >> 1         pos = pos + 1        return pos

C#

 // Returns the position of the rightmost set bit of `n` public static int positionOfRightmostSetBit(int n) {     // if the number is odd, return 1     if ((n & 1) != 0) {         return 1;     }        // unset rightmost bit and xor with the number itself     n = n ^ (n & (n - 1));        // find the position of the only set bit in the result;     // we can directly return `log2(n) + 1` from the     // function     int pos = 0;     while (n != 0) {         n = n >> 1;         pos++;     }        return pos; }

Javascript

 // Returns the position of the rightmost set bit of `n` function positionOfRightmostSetBit( n) {     // if the number is odd, return 1     if (n & 1) {         return 1;     }        // unset rightmost bit and xor with the number itself     n = n ^ (n & (n - 1));        // find the position of the only set bit in the result;     // we can directly return `log2(n) + 1` from the     // function     let pos = 0;     while (n) {         n = n >> 1;         pos++;     }     return pos; }

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

More Practice Problems on Bitwise Algorithms

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