Posted on by Kalkicode
Code Mathematics

# Bitwise Sieve

The Bitwise Sieve is a variant of the Sieve of Eratosthenes, a classical algorithm used to find all prime numbers up to a given limit. The Bitwise Sieve optimizes the space usage by representing numbers using bits in an array, which significantly reduces the memory requirement compared to a traditional array.

## Problem Statement

The problem is to efficiently find all prime numbers within a given range [2, n] using the Bitwise Sieve algorithm.

## Example

Let's take an example with n = 25.

1. Initialize an array called `sieve` with bits, where each bit corresponds to a number. A set bit indicates the number is composite (not prime), and a clear bit indicates the number is prime.
2. Start with the first prime number, 2.
3. For each prime number `p`, mark all multiples of `p` as composite starting from `p * p`, as all smaller multiples would have been marked by previous primes.
4. Repeat this process for all primes less than or equal to the square root of `n`.

## Idea to Solve

The Bitwise Sieve algorithm aims to eliminate memory overhead by representing numbers as bits in an array. Instead of using an array where each element represents a number, the algorithm uses a bit array where each bit represents a number. This way, the memory required is greatly reduced, especially when working with large ranges.

## Pseudocode

``````function non_prime(num, position):
return (num & (1 << position))

function update_status(num, position):
return (num | (1 << position))

function bitwise_sieve(n):
if n <= 1:
return

space = (n >> 5) + 2
sieve[space]

for i = 3 to sqrt(n) step 2:
slot = i >> 5
position = i & 31
if non_prime(sieve[slot], position) == 0:
for j = i * i to n step (i << 1):
slot = j >> 5
position = j & 31
sieve[slot] = update_status(sieve[slot], position)

print("Prime numbers from 2 to", n)
print("[ 2")
for i = 3 to n step 2:
slot = i >> 5
position = i & 31
if non_prime(sieve[slot], position) == 0:
print(i)
print("]")``````

## Algorithm Explanation

1. The algorithm defines two helper functions, `non_prime` and `update_status`, which operate on the bits in the sieve array.
2. The `bitwise_sieve` function initializes the sieve array and iterates through odd numbers starting from 3 up to the square root of `n`.
3. For each prime number found, it marks its multiples as non-prime in the sieve array.
4. The algorithm then prints the prime numbers in the given range.

## Code Solution

Here given code implementation process.

``````// C Program
// Print Prime numbers using
// Bitwise Sieve
#include <stdio.h>

int non_prime(int num, int position)
{
return (num & (1 << position));
}
int update_status(int num, int position)
{
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
void bitwise_sieve(int n)
{
if (n <= 1)
{
//When n are invalid to prime number
return;
}
int space = (n >> 5) + 2;
//This are used to detect prime numbers
int sieve[space];
// Loop controlling variables
int i = 0;
int j = 0;
//define some auxiliary variable
int slot = 0;
int position = 0;
for (i = 3; i * i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve[slot], position) == 0)
{
for (j = i * i; j <= n; j += (i << 1))
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve[slot] = update_status(sieve[slot], position);
}
}
}
printf("\n Prime of (2 - %d) are \n", n);
//Display first element
printf(" [ 2");
for (i = 3; i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve[slot], position) == 0)
{
//When [i] is a prime number
printf("  %d", i);
}
}
printf(" ] \n");
}
int main()
{
bitwise_sieve(25);
bitwise_sieve(101);
bitwise_sieve(200);
return 0;
}``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  11  13  17  19  23  25  29  31  35  37  41  43  47  49  53  55  59  61  67  77  79  83  85  89  91  95  97  101 ]

Prime of (2 - 200) are
[ 2  3  11  13  17  19  23  25  29  31  35  37  41  43  47  49  53  55  59  61  67  77  79  83  85  89  91  95  97  101  103  107  109  113  115  119  125  127  131  139  145  151  157  163  175  181 ]``````
``````// Java Program
// Print prime number by using
// Bitwise Sieve
class BitwiseSieve
{
public int non_prime(int num, int position)
{
return (num & (1 << position));
}
public int update_status(int num, int position)
{
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
public void bitwise_sieve(int n)
{
if (n <= 1)
{
//When n are invalid to prime number
return;
}
int space = (n >> 5) + 2;
//This are used to detect prime numbers
int[] sieve = new int[space];
// Loop controlling variables
int i = 0;
int j = 0;
//define some auxiliary variable
int slot = 0;
int position = 0;
for (i = 3; i * i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve[slot], position) == 0)
{
for (j = i * i; j <= n; j += (i << 1))
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve[slot] = update_status(sieve[slot], position);
}
}
}
System.out.print("\n Prime of (2 - " + n + ") are \n");
//Display first element
System.out.print(" [ 2");
for (i = 3; i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve[slot], position) == 0)
{
//When [i] is a prime number
System.out.print("  " + i);
}
}
System.out.print(" ] \n");
}
public static void main(String[] args)
{
BitwiseSieve obj = new BitwiseSieve();
//Test Case
obj.bitwise_sieve(25);
obj.bitwise_sieve(101);
obj.bitwise_sieve(200);
}
}``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 ]

Prime of (2 - 200) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199 ]``````
``````//Include header file
#include <iostream>
using namespace std;

// C++ Program
// Print prime number by using
// Bitwise Sieve

class BitwiseSieve
{
public:
int non_prime(int num, int position)
{
return (num & (1 << position));
}
int update_status(int num, int position)
{
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
void bitwise_sieve(int n)
{
if (n <= 1)
{
//When n are invalid to prime number
return;
}
int space = (n >> 5) + 2;
//This are used to detect prime numbers
int sieve[space];
// Loop controlling variables
int i = 0;
int j = 0;
//define some auxiliary variable
int slot = 0;
int position = 0;
for (i = 3; i *i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (this->non_prime(sieve[slot], position) == 0)
{
for (j = i *i; j <= n; j += (i << 1))
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve[slot] = this->update_status(sieve[slot], position);
}
}
}
cout << "\n Prime of (2 - " << n << ") are \n";
//Display first element
cout << " [ 2";
for (i = 3; i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (this->non_prime(sieve[slot], position) == 0)
{
//When [i] is a prime number
cout << "  " << i;
}
}
cout << " ] \n";
}
};
int main()
{
BitwiseSieve obj = BitwiseSieve();
//Test Case
obj.bitwise_sieve(25);
obj.bitwise_sieve(101);
obj.bitwise_sieve(200);
return 0;
}``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  7  11  13  23  29  47  53  55  59  61  67  71  79  83  85  89  95  97  101 ]

Prime of (2 - 200) are
[ 2  3  7  11  13  23  29  47  53  55  59  61  67  71  79  83  85  89  95  97  101  103  107  109  113  115  125  127  131  139  151  157  181  199 ]``````
``````//Include namespace system
using System;

// C# Program
// Print prime number by using
// Bitwise Sieve

class BitwiseSieve
{
public int non_prime(int num, int position)
{
return (num & (1 << position));
}
public int update_status(int num, int position)
{
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
public void bitwise_sieve(int n)
{
if (n <= 1)
{
//When n are invalid to prime number
return;
}
int space = (n >> 5) + 2;
//This are used to detect prime numbers
int[] sieve = new int[space];
// Loop controlling variables
int i = 0;
int j = 0;
//define some auxiliary variable
int slot = 0;
int position = 0;
for (i = 3; i * i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve[slot], position) == 0)
{
for (j = i * i; j <= n; j += (i << 1))
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve[slot] = update_status(sieve[slot], position);
}
}
}
Console.Write("\n Prime of (2 - " + n + ") are \n");
//Display first element
Console.Write(" [ 2");
for (i = 3; i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve[slot], position) == 0)
{
//When [i] is a prime number
Console.Write("  " + i);
}
}
Console.Write(" ] \n");
}
public static void Main(String[] args)
{
BitwiseSieve obj = new BitwiseSieve();
//Test Case
obj.bitwise_sieve(25);
obj.bitwise_sieve(101);
obj.bitwise_sieve(200);
}
}``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 ]

Prime of (2 - 200) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199 ]``````
``````<?php
// Php Program
// Print prime number by using
// Bitwise Sieve

class BitwiseSieve
{
public  function non_prime(\$num, \$position)
{
return (\$num & (1 << \$position));
}
public  function update_status(\$num, \$position)
{
return (\$num | (1 << \$position));
}
//Find all prime numbers which have smaller and equal to given number n
public  function bitwise_sieve(\$n)
{
if (\$n <= 1)
{
//When n are invalid to prime number
return;
}
\$space = (\$n >> 5) + 2;
//This are used to detect prime numbers
\$sieve = array_fill(0, \$space, 0);
// Loop controlling variables
\$i = 0;
\$j = 0;
//define some auxiliary variable
\$slot = 0;
\$position = 0;
for (\$i = 3; \$i * \$i <= \$n; \$i = \$i + 2)
{
//get slot and position
\$slot = \$i >> 5;
\$position = \$i & 31;
if (\$this->non_prime(\$sieve[\$slot], \$position) == 0)
{
for (\$j = \$i * \$i; \$j <= \$n; \$j += (\$i << 1))
{
//get slot and position
\$slot = \$j >> 5;
\$position = \$j & 31;
\$sieve[\$slot] = \$this->update_status(\$sieve[\$slot], \$position);
}
}
}
echo "\n Prime of (2 - ". \$n .") are \n";
//Display first element
echo " [ 2";
for (\$i = 3; \$i <= \$n; \$i = \$i + 2)
{
//get slot and position
\$slot = \$i >> 5;
\$position = \$i & 31;
if (\$this->non_prime(\$sieve[\$slot], \$position) == 0)
{
//When [i] is a prime number
echo "  ". \$i;
}
}
echo " ] \n";
}
}

function main()
{
\$obj = new BitwiseSieve();
//Test Case
\$obj->bitwise_sieve(25);
\$obj->bitwise_sieve(101);
\$obj->bitwise_sieve(200);
}
main();``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 ]

Prime of (2 - 200) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199 ]``````
``````// Node Js Program
// Print prime number by using
// Bitwise Sieve

class BitwiseSieve
{
non_prime(num, position)
{
return (num & (1 << position));
}
update_status(num, position)
{
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
bitwise_sieve(n)
{
if (n <= 1)
{
//When n are invalid to prime number
return;
}
var space = (n >> 5) + 2;
//This are used to detect prime numbers
var sieve = Array(space).fill(0);
// Loop controlling variables
var i = 0;
var j = 0;
//define some auxiliary variable
var slot = 0;
var position = 0;
for (i = 3; i * i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (this.non_prime(sieve[slot], position) == 0)
{
for (j = i * i; j <= n; j += (i << 1))
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve[slot] = this.update_status(sieve[slot], position);
}
}
}
process.stdout.write("\n Prime of (2 - " + n + ") are \n");
//Display first element
process.stdout.write(" [ 2");
for (i = 3; i <= n; i = i + 2)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (this.non_prime(sieve[slot], position) == 0)
{
//When [i] is a prime number
process.stdout.write("  " + i);
}
}
process.stdout.write(" ] \n");
}
}

function main()
{
var obj = new BitwiseSieve();
//Test Case
obj.bitwise_sieve(25);
obj.bitwise_sieve(101);
obj.bitwise_sieve(200);
}
main();``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 ]

Prime of (2 - 200) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199 ]``````
``````#  Python 3 Program
#  Print prime number by using
#  Bitwise Sieve

class BitwiseSieve :
def non_prime(self, num, position) :
return (num & (1 << position))

def update_status(self, num, position) :
return (num | (1 << position))

# Find all prime numbers which have smaller and equal to given number n
def bitwise_sieve(self, n) :
if (n <= 1) :
# When n are invalid to prime number
return

space = (n >> 5) + 2
# This are used to detect prime numbers
sieve = [0] * space
# define some auxiliary variable
slot = 0
position = 0
#  Loop controlling variables
i = 3
j = 0
while (i * i <= n) :
# get slot and position
slot = i >> 5
position = i & 31
if (self.non_prime(sieve[slot], position) == 0) :
j = i * i
while (j <= n) :
# get slot and position
slot = j >> 5
position = j & 31
sieve[slot] = self.update_status(sieve[slot], position)
j += (i << 1)

i = i + 2

print("\n Prime of (2 - ", n ,") are \n", end = "")
# Display first element
print(" [ 2", end = "")
i = 3
while (i <= n) :
# get slot and position
slot = i >> 5
position = i & 31
if (self.non_prime(sieve[slot], position) == 0) :
# When [i] is a prime number
print("  ", i, end = "")

i = i + 2

print(" ] \n", end = "")

def main() :
obj = BitwiseSieve()
# Test Case
obj.bitwise_sieve(25)
obj.bitwise_sieve(101)
obj.bitwise_sieve(200)

if __name__ == "__main__": main()``````

#### Output

`````` Prime of (2 -  25 ) are
[ 2   3   5   7   11   13   17   19   23 ]

Prime of (2 -  101 ) are
[ 2   3   5   7   11   13   17   19   23   29   31   37   41   43   47   53   59   61   67   71   73   79   83   89   97   101 ]

Prime of (2 -  200 ) are
[ 2   3   5   7   11   13   17   19   23   29   31   37   41   43   47   53   59   61   67   71   73   79   83   89   97   101   103   107   109   113   127   131   137   139   149   151   157   163   167   173   179   181   191   193   197   199 ]``````
``````#  Ruby Program
#  Print prime number by using
#  Bitwise Sieve
class BitwiseSieve

def non_prime(num, position)
return (num & (1 << position))
end
def update_status(num, position)

return (num | (1 << position))
end
# Find all prime numbers which have smaller and equal to given number n
def bitwise_sieve(n)

if (n <= 1)
# When n are invalid to prime number
return
end
space = (n >> 5) + 2
# This are used to detect prime numbers
sieve = Array.new(space) {0}
# define some auxiliary variable
slot = 0
position = 0
#  Loop controlling variables
i = 3
j = 0
while (i * i <= n)

# get slot and position
slot = i >> 5
position = i & 31
if (self.non_prime(sieve[slot], position) == 0)

j = i * i
while (j <= n)

# get slot and position
slot = j >> 5
position = j & 31
sieve[slot] = self.update_status(sieve[slot], position)
j += (i << 1)
end
end
i = i + 2
end
print("\n Prime of (2 - ", n ,") are \n")
# Display first element
print(" [ 2")
i = 3
while (i <= n)

# get slot and position
slot = i >> 5
position = i & 31
if (self.non_prime(sieve[slot], position) == 0)

# When [i] is a prime number
print("  ", i)
end
i = i + 2
end
print(" ] \n")
end
end
def main()

obj = BitwiseSieve.new()
# Test Case
obj.bitwise_sieve(25)
obj.bitwise_sieve(101)
obj.bitwise_sieve(200)
end
main()``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 ]

Prime of (2 - 200) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199 ]
``````
``````// Scala Program
// Print prime number by using
// Bitwise Sieve
class BitwiseSieve
{
def non_prime(num: Int, position: Int): Int = {
return (num & (1 << position));
}
def update_status(num: Int, position: Int): Int = {
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
def bitwise_sieve(n: Int): Unit = {
if (n <= 1)
{
//When n are invalid to prime number
return;
}
var space: Int = (n >> 5) + 2;
//This are used to detect prime numbers
var sieve: Array[Int] = Array.fill[Int](space)(0);
//define some auxiliary variable
var slot: Int = 0;
var position: Int = 0;
// Loop controlling variables
var i: Int = 3;
var j: Int = 0;
while (i * i <= n)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve(slot), position) == 0)
{
j = i * i;
while (j <= n)
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve(slot) = update_status(sieve(slot), position);
j += (i << 1);
}
}
i = i + 2;
}
print("\n Prime of (2 - " + n + ") are \n");
//Display first element
print(" [ 2");
i = 3;
while (i <= n)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (non_prime(sieve(slot), position) == 0)
{
//When [i] is a prime number
print("  " + i);
}
i = i + 2;
}
print(" ] \n");
}
}
object Main
{
def main(args: Array[String]): Unit = {
var obj: BitwiseSieve = new BitwiseSieve();
//Test Case
obj.bitwise_sieve(25);
obj.bitwise_sieve(101);
obj.bitwise_sieve(200);
}
}``````

#### Output

`````` Prime of (2 - 25) are
[ 2  3  5  7  11  13  17  19  23 ]

Prime of (2 - 101) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101 ]

Prime of (2 - 200) are
[ 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199 ]``````
``````// Swift Program
// Print prime number by using
// Bitwise Sieve
class BitwiseSieve
{
func non_prime(_ num: Int, _ position: Int) -> Int
{
return (num & (1 << position));
}
func update_status(_ num: Int, _ position: Int) -> Int
{
return (num | (1 << position));
}
//Find all prime numbers which have smaller and equal to given number n
func bitwise_sieve(_ n: Int)
{
if (n <= 1)
{
//When n are invalid to prime number
return;
}
let space: Int = (n >> 5) + 2;
//This are used to detect prime numbers
var sieve: [Int] = Array(repeating: 0, count: space);
//define some auxiliary variable
var slot: Int = 0;
var position: Int = 0;
// Loop controlling variables
var i: Int = 3;
var j: Int = 0;
while (i * i <= n)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (self.non_prime(sieve[slot], position) == 0)
{
j = i * i;
while (j <= n)
{
//get slot and position
slot = j >> 5;
position = j & 31;
sieve[slot] = self.update_status(sieve[slot], position);
j += (i << 1);
}
}
i = i + 2;
}
print("\n Prime of (2 - ", n ,") are \n", terminator: "");
//Display first element
print(" [ 2", terminator: "");
i = 3;
while (i <= n)
{
//get slot and position
slot = i >> 5;
position = i & 31;
if (self.non_prime(sieve[slot], position) == 0)
{
//When [i] is a prime number
print("  ", i, terminator: "");
}
i = i + 2;
}
print(" ] \n", terminator: "");
}
}
func main()
{
let obj: BitwiseSieve = BitwiseSieve();
//Test Case
obj.bitwise_sieve(25);
obj.bitwise_sieve(101);
obj.bitwise_sieve(200);
}
main();``````

#### Output

`````` Prime of (2 -  25 ) are
[ 2   3   5   7   11   13   17   19   23 ]

Prime of (2 -  101 ) are
[ 2   3   5   7   11   13   17   19   23   29   31   37   41   43   47   53   59   61   67   71   73   79   83   89   97   101 ]

Prime of (2 -  200 ) are
[ 2   3   5   7   11   13   17   19   23   29   31   37   41   43   47   53   59   61   67   71   73   79   83   89   97   101   103   107   109   113   127   131   137   139   149   151   157   163   167   173   179   181   191   193   197   199 ]``````

## Resultant Output Explanation

The output displays the prime numbers within the specified ranges. For example, for `bitwise_sieve(25)`, the prime numbers from 2 to 25 are [2, 3, 5, 7, 11, 13, 17, 19, 23].

## Time Complexity

The time complexity of the Bitwise Sieve algorithm is O(n log log n), where n is the upper limit of the range. This is because the algorithm eliminates the multiples of each prime number up to the square root of n. The space complexity is reduced to O(n/32), which represents the size of the sieve array in terms of bits.

## Comment

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