Posted on by Kalkicode
Code Mathematics

# Find the prime numbers between given range using segmented sieve

The segmented sieve is an optimization of the Sieve of Eratosthenes algorithm that allows us to find prime numbers in a specific range efficiently. The segmented sieve divides the range into segments and applies the sieve only to those segments, rather than the entire range.

## Problem Statement

The problem is to find and print all prime numbers within a given range [s, e], where s and e are positive integers.

## Idea to Solve the Problem

To solve this problem using the segmented sieve, we can follow these steps:

1. Use the regular Sieve of Eratosthenes to find all prime numbers up to the square root of the upper limit (e).

2. Initialize two pointers `low` and `high` to the starting range s and the next segment's limit (s + √e).

3. Create a boolean array `mark` of size `limit + 1` (where `limit` is √e) to mark the sieve values within the current segment.

4. Loop through the prime numbers obtained from step 1. For each prime `p`, calculate the value `value = (low / p) * p`, and set it to the next multiple of `p` within the current segment. Mark all multiples of `p` as non-prime in the `mark` array.

5. After marking the multiples of primes in the current segment, print the prime numbers in the segment.

6. Update `low` and `high` to the next segment's limits and repeat steps 3 to 5 until the entire range [s, e] is covered.

## Pseudocode

``````class Sieve
method eratosthenesSieve(prime, n)
mark = boolean array of size n + 1
for i from 0 to n
mark[i] = true
mark[0] = false
mark[1] = false
for i from 2 to n
if mark[i] is true
for j from i * i to n step i
mark[j] = false

method segmentedSieve(s, e)
if s < 0 or e < 2
return

prime = empty array of integers
limit = floor(sqrt(e)) + 1
low = s
high = limit + s
mark = boolean array of size limit + 1

eratosthenesSieve(prime, limit)

loop until low < e
for i from 0 to limit
mark[i] = true
if high >= e
high = e
for i from 0 to prime.size
value = floor(low / prime[i]) * prime[i]
if value < low
value += prime[i]
for j from value to high step prime[i]
mark[j - low] = false
for i from low to high
if mark[i - low] is true
print i
high = high + limit
low = low + limit

function main
task.segmentedSieve(100, 200) // Test case A
task.segmentedSieve(999, 1200) // Test case B``````

## Algorithm Explanation

1. The `eratosthenesSieve` method implements the Sieve of Eratosthenes to find all prime numbers up to `n`.

2. The `segmentedSieve` method finds and prints prime numbers within the range [s, e] using the segmented sieve algorithm:

• It first checks if the input values are valid.
• It initializes the variables and arrays required for the algorithm.
• It uses the `eratosthenesSieve` method to find the prime numbers up to √e.
• It then loops through the segments, marking non-prime values and printing prime numbers within each segment.

## Code Solution

``````import java.util.ArrayList;
// Java program for
// Find the prime numbers between given range using segmented sieve
public class Sieve
{
public void eratosthenesSieve(ArrayList < Integer > prime, int n)
{
boolean[] mark = new boolean[n + 1];
// Set all element as prime
for (int i = 0; i <= n; ++i)
{
mark[i] = true;
}
mark[0] = false;
mark[1] = false;
for (int i = 2; i <= n; ++i)
{
if (mark[i] == true)
{
// Collect prime element
for (int j = i * i; j <= n; j += i)
{
mark[j] = false;
}
}
}
}
public void segmentedSieve(int s, int e)
{
if (s < 0 || e < 2)
{
return;
}
System.out.println("\n Prime number in range of (" + s + "," + e + ")");
ArrayList < Integer > prime = new ArrayList < Integer > ();
// Get the initial prime number by given e
int limit = (int)(Math.floor(Math.sqrt(e)) + 1);
// Starting value
int low = s;
int high = (limit) + s;
int value = 0;
// Container which is used to detect (√e) prime element
boolean[] mark = new boolean[limit + 1];
// Find first (√e) prime number
eratosthenesSieve(prime, limit);
for (int i = 0; i < prime.size(); ++i)
{
if (prime.get(i) >= s)
{
System.out.print("  " + prime.get(i));
}
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
// Set next (√e) prime number is valid
for (int i = 0; i <= limit; ++i)
{
mark[i] = true;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
for (int i = 0; i < prime.size(); i++)
{
value = (int)(Math.floor(low / prime.get(i)) * prime.get(i));
if (value < low)
{
value += prime.get(i);
}
for (int j = value; j < high; j += prime.get(i))
{
// Set mutiple is non prime
mark[j - low] = false;
}
}
// Display prime elements
for (int i = low; i < high; i++)
{
if (mark[i - low] == true)
{
System.out.print("  " + i);
}
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
public static void main(String[] args)
{
// Test
}
}``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````// Include header file
#include <iostream>
#include <vector>
#include <math.h>

using namespace std;
// C++ program for
// Find the prime numbers between given range using segmented sieve
class Sieve
{
public: void eratosthenesSieve(vector < int > &prime, int n)
{
bool mark[n + 1];
// Set all element as prime
for (int i = 0; i <= n; ++i)
{
mark[i] = true;
}
mark[0] = false;
mark[1] = false;
for (int i = 2; i <= n; ++i)
{
if (mark[i] == true)
{
// Collect prime element
prime.push_back(i);
for (int j = i *i; j <= n; j += i)
{
mark[j] = false;
}
}
}
}
void segmentedSieve(int s, int e)
{
if (s < 0 || e < 2)
{
return;
}
cout << "\n Prime number in range of ("
<< s << "," << e << ")" << endl;
vector < int > prime;
// Get the initial prime number by given e
int limit = (int)(floor(sqrt(e)) + 1);
// Starting value
int low = s;
int high = limit + s;
int value = 0;
// Container which is used to detect (√e) prime element
bool mark[limit + 1];
// Find first (√e) prime number
this->eratosthenesSieve(prime, limit);
for (int i = 0; i < prime.size(); ++i)
{
if (prime.at(i) >= s)
{
cout << "  " << prime.at(i);
}
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
// Set next (√e) prime number is valid
for (int i = 0; i <= limit; ++i)
{
mark[i] = true;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
for (int i = 0; i < prime.size(); i++)
{
value = (int)(floor(low / prime.at(i)) * prime.at(i));
if (value < low)
{
value += prime.at(i);
}
for (int j = value; j < high; j += prime.at(i))
{
// Set mutiple is non prime
mark[j - low] = false;
}
}
// Display prime elements
for (int i = low; i < high; i++)
{
if (mark[i - low] == true)
{
cout << "  " << i;
}
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
};
int main()
{
// Test
return 0;
}``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````// Include namespace system
using System;
using System.Collections.Generic;
// Csharp program for
// Find the prime numbers between given range using segmented sieve
public class Sieve
{
public void eratosthenesSieve(List < int > prime, int n)
{
Boolean[] mark = new Boolean[n + 1];
// Set all element as prime
for (int i = 0; i <= n; ++i)
{
mark[i] = true;
}
mark[0] = false;
mark[1] = false;
for (int i = 2; i <= n; ++i)
{
if (mark[i] == true)
{
// Collect prime element
for (int j = i * i; j <= n; j += i)
{
mark[j] = false;
}
}
}
}
public void segmentedSieve(int s, int e)
{
if (s < 0 || e < 2)
{
return;
}
Console.WriteLine("\n Prime number in range of (" + s + "," + e + ")");
List < int > prime = new List < int > ();
// Get the initial prime number by given e
int limit = (int)(Math.Floor(Math.Sqrt(e)) + 1);
// Starting value
int low = s;
int high = (limit) + s;
int value = 0;
// Container which is used to detect (√e) prime element
Boolean[] mark = new Boolean[limit + 1];
// Find first (√e) prime number
this.eratosthenesSieve(prime, limit);
for (int i = 0; i < prime.Count; ++i)
{
if (prime[i] >= s)
{
Console.Write("  " + prime[i]);
}
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
// Set next (√e) prime number is valid
for (int i = 0; i <= limit; ++i)
{
mark[i] = true;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
for (int i = 0; i < prime.Count; i++)
{
value = (int)(Math.Floor((double)(low / prime[i])) * prime[i]);
if (value < low)
{
value += prime[i];
}
for (int j = value; j < high; j += prime[i])
{
// Set mutiple is non prime
mark[j - low] = false;
}
}
// Display prime elements
for (int i = low; i < high; i++)
{
if (mark[i - low] == true)
{
Console.Write("  " + i);
}
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
public static void Main(String[] args)
{
// Test
}
}``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````package main
import "math"
import "fmt"
// Go program for
// Find the prime numbers between given range using segmented sieve
type Sieve struct {}
func getSieve() * Sieve {
var me *Sieve = &Sieve {}
return me
}
func(this Sieve) eratosthenesSieve(prime *[]int, n int) {
var mark = make([] bool, n + 1)
// Set all element as prime
for i := 0 ; i <= n ; i++ {
mark[i] = true
}
mark[0] = false
mark[1] = false
for i := 2 ; i <= n ; i++ {
if mark[i] == true {
// Collect prime element
*prime = append(*prime, i)
for j := i * i ; j <= n ; j += i {
mark[j] = false
}
}
}
}
func(this Sieve) segmentedSieve(s, e int) {
if s < 0 || e < 2 {
return
}
fmt.Println("\n Prime number in range of (", s, ",", e, ")")
var prime = make([]int ,0)
// Get the initial prime number by given e
var limit int = (int)(math.Floor(math.Sqrt(float64(e))) + 1)
// Starting value
var low int = s
var high int = (limit) + s
var value int = 0
// Container which is used to detect (√e) prime element
var mark = make([] bool, limit + 1)
// Find first (√e) prime number
this.eratosthenesSieve(&prime, limit)
for i := 0 ; i < len(prime) ; i++ {
if prime[i] >= s {
fmt.Print("  ", prime[i])
}
}
// This loop displays the remaining prime number between (√e .. e)
for (low < e) {
// Set next (√e) prime number is valid
for i := 0 ; i <= limit ; i++ {
mark[i] = true
}
if high >= e {
// When next prime pair are greater than e
// Set high value to e
high = e
}
for i := 0 ; i < len(prime) ; i++ {
value = (int)(math.Floor(float64(low / prime[i])) * float64(prime[i]))
if value < low {
value += prime[i]
}
for j := value ; j < high ; j += prime[i] {
// Set mutiple is non prime
mark[j - low] = false
}
}
// Display prime elements
for i := low ; i < high ; i++ {
if mark[i - low] == true {
fmt.Print("  ", i)
}
}
// Update of all multiple of value is non prime
high = high + limit
low = low + limit
}
}
func main() {
var task * Sieve = getSieve()
// Test
}``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````<?php
// Php program for
// Find the prime numbers between given range using segmented sieve
class Sieve
{
public	function eratosthenesSieve(&\$prime, \$n)
{
// Set all element as prime
\$mark = array_fill(0, \$n + 1, true);
\$mark[0] = false;
\$mark[1] = false;
for (\$i = 2; \$i <= \$n; ++\$i)
{
if (\$mark[\$i] == true)
{
// Collect prime element
\$prime[] = \$i;
for (\$j = \$i * \$i; \$j <= \$n; \$j += \$i)
{
\$mark[\$j] = false;
}
}
}
}
public	function segmentedSieve(\$s, \$e)
{
if (\$s < 0 || \$e < 2)
{
return;
}
echo("\n Prime number in range of (".\$s.
",".\$e.
")".
"\n");
\$prime = array();
// Get the initial prime number by given e
\$limit = (int)(floor(sqrt(\$e)) + 1);
// Starting value
\$low = \$s;
\$high = (\$limit) + \$s;
\$value = 0;
// Container which is used to detect (√e) prime element
\$mark = array_fill(0, \$limit + 1, false);
// Find first (√e) prime number
\$this->eratosthenesSieve(\$prime, \$limit);
for (\$i = 0; \$i < count(\$prime); ++\$i)
{
if (\$prime[\$i] >= \$s)
{
echo("  ".\$prime[\$i]);
}
}
// This loop displays the remaining prime number between (√e .. e)
while (\$low < \$e)
{
// Set next (√e) prime number is valid
for (\$i = 0; \$i <= \$limit; ++\$i)
{
\$mark[\$i] = true;
}
if (\$high >= \$e)
{
// When next prime pair are greater than e
// Set high value to e
\$high = \$e;
}
for (\$i = 0; \$i < count(\$prime); \$i++)
{
\$value = (int)(floor((int)(\$low / \$prime[\$i])) * \$prime[\$i]);
if (\$value < \$low)
{
\$value += \$prime[\$i];
}
for (\$j = \$value; \$j < \$high; \$j += \$prime[\$i])
{
// Set mutiple is non prime
\$mark[\$j - \$low] = false;
}
}
// Display prime elements
for (\$i = \$low; \$i < \$high; \$i++)
{
if (\$mark[\$i - \$low] == true)
{
echo("  ".\$i);
}
}
// Update of all multiple of value is non prime
\$high = \$high + \$limit;
\$low = \$low + \$limit;
}
}
}

function main()
{
// Test
}
main();``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````// Node JS program for
// Find the prime numbers between given range using segmented sieve
class Sieve
{
eratosthenesSieve(prime, n)
{
// Set all element as prime
var mark = Array(n + 1).fill(true);
mark[0] = false;
mark[1] = false;
for (var i = 2; i <= n; ++i)
{
if (mark[i] == true)
{
// Collect prime element
prime.push(i);
for (var j = i * i; j <= n; j += i)
{
mark[j] = false;
}
}
}
}
segmentedSieve(s, e)
{
if (s < 0 || e < 2)
{
return;
}
console.log("\n Prime number in range of (" +
s + "," + e + ")");
var prime = [];
// Get the initial prime number by given e
var limit = parseInt(Math.floor(Math.sqrt(e)) + 1);
// Starting value
var low = s;
var high = (limit) + s;
var value = 0;
// Container which is used to detect (√e) prime element
var mark = Array(limit + 1).fill(false);
// Find first (√e) prime number
this.eratosthenesSieve(prime, limit);
for (var i = 0; i < prime.length; ++i)
{
if (prime[i] >= s)
{
process.stdout.write("  " + prime[i]);
}
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
// Set next (√e) prime number is valid
for (var i = 0; i <= limit; ++i)
{
mark[i] = true;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
for (var i = 0; i < prime.length; i++)
{
value = parseInt(
Math.floor(parseInt(low / prime[i])) * prime[i]);
if (value < low)
{
value += prime[i];
}
for (var j = value; j < high; j += prime[i])
{
// Set mutiple is non prime
mark[j - low] = false;
}
}
// Display prime elements
for (var i = low; i < high; i++)
{
if (mark[i - low] == true)
{
process.stdout.write("  " + i);
}
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
}

function main()
{
// Test
}
main();``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````import math
#  Python 3 program for
#  Find the prime numbers between given range using segmented sieve
class Sieve :
def eratosthenesSieve(self, prime, n) :
#  Set all element as prime
mark = [True] * (n + 1)
mark[0] = False
mark[1] = False
i = 2
while (i <= n) :
if (mark[i] == True) :
#  Collect prime element
prime.append(i)
j = i * i
while (j <= n) :
mark[j] = False
j += i

i += 1

def segmentedSieve(self, s, e) :
if (s < 0 or e < 2) :
return

print("\n Prime number in range of (", s ,",", e ,")")
prime = []
#  Get the initial prime number by given e
limit = (int)(math.floor(math.sqrt(e)) + 1)
#  Starting value
low = s
high = (limit) + s
value = 0
#  Container which is used to detect (√e) prime element
mark = [False] * (limit + 1)
#  Find first (√e) prime number
self.eratosthenesSieve(prime, limit)
i = 0
while (i < len(prime)) :
if (prime[i] >= s) :
print("  ", prime[i], end = "")

i += 1

#  This loop displays the remaining prime number between (√e .. e)
while (low < e) :
i = 0
#  Set next (√e) prime number is valid
while (i <= limit) :
mark[i] = True
i += 1

if (high >= e) :
#  When next prime pair are greater than e
#  Set high value to e
high = e

i = 0
while (i < len(prime)) :
value = (int)(math.floor(int(low / prime[i])) * prime[i])
if (value < low) :
value += prime[i]

j = value
while (j < high) :
#  Set mutiple is non prime
mark[j - low] = False
j += prime[i]

i += 1

i = low
#  Display prime elements
while (i < high) :
if (mark[i - low] == True) :
print("  ", i, end = "")

i += 1

#  Update of all multiple of value is non prime
high = high + limit
low = low + limit

def main() :
#  Test

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

#### Output

`````` Prime number in range of ( 100 , 200 )
101   103   107   109   113   127   131   137   139   149   151   157   163   167   173   179   181   191   193   197   199
Prime number in range of ( 999 , 1200 )
1009   1013   1019   1021   1031   1033   1039   1049   1051   1061   1063   1069   1087   1091   1093   1097   1103   1109   1117   1123   1129   1151   1153   1163   1171   1181   1187   1193``````
``````#  Ruby program for
#  Find the prime numbers between given range using segmented sieve
class Sieve
def eratosthenesSieve(prime, n)
#  Set all element as prime
mark = Array.new(n + 1) {true}
mark[0] = false
mark[1] = false
i = 2
while (i <= n)
if (mark[i] == true)
#  Collect prime element
prime.push(i)
j = i * i
while (j <= n)
mark[j] = false
j += i
end

end

i += 1
end

end

def segmentedSieve(s, e)
if (s < 0 || e < 2)
return
end

print("\n Prime number in range of (", s ,",", e ,")", "\n")
prime = []
#  Get the initial prime number by given e
limit = (Math.sqrt(e).floor() + 1)
#  Starting value
low = s
high = (limit) + s
value = 0
#  Container which is used to detect (√e) prime element
mark = Array.new(limit + 1) {false}
#  Find first (√e) prime number
self.eratosthenesSieve(prime, limit)
i = 0
while (i < prime.length)
if (prime[i] >= s)
print("  ", prime[i])
end

i += 1
end

#  This loop displays the remaining prime number between (√e .. e)
while (low < e)
i = 0
#  Set next (√e) prime number is valid
while (i <= limit)
mark[i] = true
i += 1
end

if (high >= e)
#  When next prime pair are greater than e
#  Set high value to e
high = e
end

i = 0
while (i < prime.length)
value = (low / prime[i].floor() * prime[i])
if (value < low)
value += prime[i]
end

j = value
while (j < high)
#  Set mutiple is non prime
mark[j - low] = false
j += prime[i]
end

i += 1
end

i = low
#  Display prime elements
while (i < high)
if (mark[i - low] == true)
print("  ", i)
end

i += 1
end

#  Update of all multiple of value is non prime
high = high + limit
low = low + limit
end

end

end

def main()
#  Test
end

main()``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````import scala.collection.mutable._;
// Scala program for
// Find the prime numbers between given range using segmented sieve
class Sieve()
{
def eratosthenesSieve(prime: ArrayBuffer[Int], n: Int): Unit = {
// Set all element as prime
var mark: Array[Boolean] = Array.fill[Boolean](n + 1)(true);
mark(0) = false;
mark(1) = false;
var i: Int = 2;
while (i <= n)
{
if (mark(i) == true)
{
// Collect prime element
prime += i;
var j: Int = i * i;
while (j <= n)
{
mark(j) = false;
j += i;
}
}
i += 1;
}
}
def segmentedSieve(s: Int, e: Int): Unit = {
if (s < 0 || e < 2)
{
return;
}
println("\n Prime number in range of (" + s + "," + e + ")");
var prime: ArrayBuffer[Int] = new ArrayBuffer[Int]();
// Get the initial prime number by given e
var limit: Int = (Math.floor(scala.math.sqrt(e)) + 1).toInt;
// Starting value
var low: Int = s;
var high: Int = (limit) + s;
var value: Int = 0;
// Container which is used to detect (√e) prime element
var mark: Array[Boolean] = Array.fill[Boolean](limit + 1)(false);
// Find first (√e) prime number
eratosthenesSieve(prime, limit);
var i: Int = 0;
while (i < prime.size)
{
if (prime(i) >= s)
{
print("  " + prime(i));
}
i += 1;
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
i = 0;
// Set next (√e) prime number is valid
while (i <= limit)
{
mark(i) = true;
i += 1;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
i = 0;
while (i < prime.size)
{
value = (Math.floor(low / prime(i)) * prime(i)).toInt;
if (value < low)
{
value += prime(i);
}
var j: Int = value;
while (j < high)
{
// Set mutiple is non prime
mark(j - low) = false;
j += prime(i);
}
i += 1;
}
i = low;
// Display prime elements
while (i < high)
{
if (mark(i - low) == true)
{
print("  " + i);
}
i += 1;
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
}
object Main
{
def main(args: Array[String]): Unit = {
var task: Sieve = new Sieve();
// Test
}
}``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````
``````import Foundation;
// Swift 4 program for
// Find the prime numbers between given range using segmented sieve
class Sieve
{
func eratosthenesSieve(_ prime: inout[Int], _ n: Int)
{
// Set all element as prime
var mark: [Bool] = Array(repeating: true, count: n + 1);
mark[0] = false;
mark[1] = false;
var i: Int = 2;
while (i <= n)
{
if (mark[i] == true)
{
// Collect prime element
prime.append(i);
var j: Int = i * i;
while (j <= n)
{
mark[j] = false;
j += i;
}
}
i += 1;
}
}
func segmentedSieve(_ s: Int, _ e: Int)
{
if (s < 0 || e < 2)
{
return;
}
print("\n Prime number in range of (", s ,",", e ,")");
var prime: [Int] = [Int]();
// Get the initial prime number by given e
let limit: Int = Int((floor(Double(e).squareRoot()) + 1));
// Starting value
var low: Int = s;
var high: Int = (limit) + s;
var value: Int = 0;
// Container which is used to detect (√e) prime element
var mark: [Bool] = Array(repeating: false, count: limit + 1);
// Find first (√e) prime number
self.eratosthenesSieve(&prime, limit);
var i: Int = 0;
while (i < prime.count)
{
if (prime[i] >= s)
{
print("  ", prime[i], terminator: "");
}
i += 1;
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
i = 0;
// Set next (√e) prime number is valid
while (i <= limit)
{
mark[i] = true;
i += 1;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
i = 0;
while (i < prime.count)
{
value = Int((floor(Double(low / prime[i])) * Double(prime[i])));
if (value < low)
{
value += prime[i];
}
var j: Int = value;
while (j < high)
{
// Set mutiple is non prime
mark[j - low] = false;
j += prime[i];
}
i += 1;
}
i = low;
// Display prime elements
while (i < high)
{
if (mark[i - low] == true)
{
print("  ", i, terminator: "");
}
i += 1;
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
}
func main()
{
// Test
}
main();``````

#### Output

`````` Prime number in range of ( 100 , 200 )
101   103   107   109   113   127   131   137   139   149   151   157   163   167   173   179   181   191   193   197   199
Prime number in range of ( 999 , 1200 )
1009   1013   1019   1021   1031   1033   1039   1049   1051   1061   1063   1069   1087   1091   1093   1097   1103   1109   1117   1123   1129   1151   1153   1163   1171   1181   1187   1193``````
``````// Kotlin program for
// Find the prime numbers between given range using segmented sieve
class Sieve
{
fun eratosthenesSieve(prime: MutableList < Int >  , n : Int): Unit
{
// Set all element as prime
val mark: Array < Boolean > = Array(n + 1)
{
true
};
mark[0] = false;
mark[1] = false;
var i: Int = 2;
while (i <= n)
{
if (mark[i] == true)
{
// Collect prime element
var j: Int = i * i;
while (j <= n)
{
mark[j] = false;
j += i;
}
}
i += 1;
}
}
fun segmentedSieve(s: Int, e: Int): Unit
{
if (s < 0 || e < 2)
{
return;
}
println("\n Prime number in range of (" + s + "," + e + ")");
var prime: MutableList < Int > = mutableListOf < Int > ();
// Get the initial prime number by given e
val limit: Int = (Math.floor(Math.sqrt(e.toDouble())) + 1.0).toInt();
// Starting value
var low: Int = s;
var high: Int = (limit) + s;
var value: Int ;
// Container which is used to detect (√e) prime element
val mark: Array < Boolean > = Array(limit + 1)
{
false
};
// Find first (√e) prime number
this.eratosthenesSieve(prime, limit);
var i: Int = 0;
while (i < prime.size)
{
if (prime[i] >= s)
{
print("  " + prime[i]);
}
i += 1;
}
// This loop displays the remaining prime number between (√e .. e)
while (low < e)
{
i = 0;
// Set next (√e) prime number is valid
while (i <= limit)
{
mark[i] = true;
i += 1;
}
if (high >= e)
{
// When next prime pair are greater than e
// Set high value to e
high = e;
}
i = 0;
while (i < prime.size)
{
value = (Math.floor(
(low / prime[i]).toDouble()
) * prime[i]).toInt();
if (value < low)
{
value += prime[i];
}
var j: Int = value;
while (j < high)
{
// Set mutiple is non prime
mark[j - low] = false;
j += prime[i];
}
i += 1;
}
i = low;
// Display prime elements
while (i < high)
{
if (mark[i - low] == true)
{
print("  " + i);
}
i += 1;
}
// Update of all multiple of value is non prime
high = high + limit;
low = low + limit;
}
}
}
fun main(args: Array < String > ): Unit
{
// Test
}``````

#### Output

`````` Prime number in range of (100,200)
101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181  191  193  197  199
Prime number in range of (999,1200)
1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091  1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193``````

## Time Complexity

• Generating prime numbers using the Sieve of Eratosthenes has a time complexity of O(n log log n).
• The segmented sieve algorithm has a time complexity of O((e - s + 1) * log log e), where e is the upper limit and s is the lower limit of the range.

## Comment

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