Sieve of Eratosthenes in 0(n) time complexity - GeeksforGeeks

12 Nov, 2021
The classical Sieve of Eratosthenes algorithm takes O(N log (log N)) time to find all prime numbers less than N. In this article, a modified Sieve is discussed that works in O(N) time.Example : 
Given a number N, print all prime 
numbers smaller than N

Input :  int N = 15
Output : 2 3 5 7 11 13

Input : int N = 20
Output : 2 3 5 7 11 13 17 19
Manipulated Sieve of Eratosthenes algorithm works as follows:  
For every number i where i varies from 2 to N-1:
    Check if the number is prime. If the number
    is prime, store it in prime array.

For every prime numbers j less than or equal to the smallest  
prime factor p of i:
    Mark all numbers i*p as non_prime.
    Mark smallest prime factor of i*p as j
Below is the implementation of the above idea.  
C++
Java
Python3
PHP
Javascript
// C++ program to generate all prime numbers// less than N in O(N)#include<bits/stdc++.h>using namespace std;const long long MAX_SIZE = 1000001; // isPrime[] : isPrime[i] is true if number is prime// prime[] : stores all prime number less than N// SPF[] that store smallest prime factor of number// [for Exp : smallest prime factor of '8' and '16'// is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]vector<long long >isprime(MAX_SIZE , true);vector<long long >prime;vector<long long >SPF(MAX_SIZE); // function generate all prime number less then N in O(n)void manipulated_seive(int N){    // 0 and 1 are not prime    isprime[0] = isprime[1] = false ;     // Fill rest of the entries    for (long long int i=2; i<N ; i++)    {        // If isPrime[i] == True then i is        // prime number        if (isprime[i])        {            // put i into prime[] vector            prime.push_back(i);             // A prime number is its own smallest            // prime factor            SPF[i] = i;        }         // Remove all multiples of  i*prime[j] which are        // not prime by making isPrime[i*prime[j]] = false        // and put smallest prime factor of i*Prime[j] as prime[j]        // [ for exp :let  i = 5 , j = 0 , prime[j] = 2 [ i*prime[j] = 10 ]        // so smallest prime factor of '10' is '2' that is prime[j] ]        // this loop run only one time for number which are not prime        for (long long int j=0;             j < (int)prime.size() &&             i*prime[j] < N && prime[j] <= SPF[i];             j++)        {            isprime[i*prime[j]]=false;             // put smallest prime factor of i*prime[j]            SPF[i*prime[j]] = prime[j] ;        }    }} // driver  program to test above functionint main(){    int N = 13 ; // Must be less than MAX_SIZE     manipulated_seive(N);     // print all prime number less then N    for (int i=0; i<prime.size() && prime[i] <= N ; i++)        cout << prime[i] << " ";     return 0;}
// Java program to generate all prime numbers// less than N in O(N)  import java.util.Vector; class Test{    static final int MAX_SIZE = 1000001;    // isPrime[] : isPrime[i] is true if number is prime    // prime[] : stores all prime number less than N    // SPF[] that store smallest prime factor of number    // [for Exp : smallest prime factor of '8' and '16'    // is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]    static Vector<Boolean>isprime = new Vector<>(MAX_SIZE);    static Vector<Integer>prime = new Vector<>();    static Vector<Integer>SPF = new Vector<>(MAX_SIZE);          // method generate all prime number less then N in O(n)    static void manipulated_seive(int N)    {        // 0 and 1 are not prime        isprime.set(0, false);        isprime.set(1, false);                 // Fill rest of the entries        for (int i=2; i<N ; i++)        {            // If isPrime[i] == True then i is            // prime number            if (isprime.get(i))            {                // put i into prime[] vector                prime.add(i);                      // A prime number is its own smallest                // prime factor                SPF.set(i,i);            }                  // Remove all multiples of  i*prime[j] which are            // not prime by making isPrime[i*prime[j]] = false            // and put smallest prime factor of i*Prime[j] as prime[j]            // [for exp :let  i = 5, j = 0, prime[j] = 2 [ i*prime[j] = 10]            // so smallest prime factor of '10' is '2' that is prime[j] ]            // this loop run only one time for number which are not prime            for (int j=0;                 j < prime.size() &&                 i*prime.get(j) < N && prime.get(j) <= SPF.get(i);                 j++)            {                isprime.set(i*prime.get(j),false);                      // put smallest prime factor of i*prime[j]                SPF.set(i*prime.get(j),prime.get(j)) ;            }        }    }         // Driver method    public static void main(String args[])    {        int N = 13 ; // Must be less than MAX_SIZE                 // initializing isprime and spf        for (int i = 0; i < MAX_SIZE; i++){            isprime.add(true);            SPF.add(2);        }                  manipulated_seive(N);              // print all prime number less then N        for (int i=0; i<prime.size() && prime.get(i) <= N ; i++)            System.out.print(prime.get(i) + " ");    }}
# Python3 program to generate all# prime numbers less than N in O(N) MAX_SIZE = 1000001 # isPrime[] : isPrime[i] is true if#             number is prime# prime[] : stores all prime number#           less than N# SPF[] that store smallest prime# factor of number [for ex : smallest# prime factor of '8' and '16'# is '2' so we put SPF[8] = 2 ,# SPF[16] = 2 ]isprime = [True] * MAX_SIZEprime = []SPF = [None] * (MAX_SIZE) # function generate all prime number# less then N in O(n)def manipulated_seive(N):     # 0 and 1 are not prime    isprime[0] = isprime[1] = False     # Fill rest of the entries    for i in range(2, N):             # If isPrime[i] == True then i is        # prime number        if isprime[i] == True:                     # put i into prime[] vector            prime.append(i)             # A prime number is its own smallest            # prime factor            SPF[i] = i                 # Remove all multiples of i*prime[j]        # which are not prime by making is        # Prime[i * prime[j]] = false and put        # smallest prime factor of i*Prime[j]        # as prime[j] [ for exp :let i = 5 , j = 0 ,        # prime[j] = 2 [ i*prime[j] = 10 ]        # so smallest prime factor of '10' is '2'        # that is prime[j] ] this loop run only one        # time for number which are not prime        j = 0        while (j < len(prime) and               i * prime[j] < N and                   prime[j] <= SPF[i]):                     isprime[i * prime[j]] = False             # put smallest prime factor of i*prime[j]            SPF[i * prime[j]] = prime[j]                         j += 1         # Driver Codeif __name__ == "__main__":     N = 13 # Must be less than MAX_SIZE     manipulated_seive(N)     # print all prime number less then N    i = 0    while i < len(prime) and prime[i] <= N:        print(prime[i], end = " ")        i += 1         # This code is contributed by Rituraj Jain
<?php// PHP program to generate all// prime numbers less than N in O(N) $MAX_SIZE = 10001; // isPrime[] : isPrime[i] is true if//               number is prime// prime[] : stores all prime number//             less than N// SPF[] that store smallest prime// factor of number [for ex : smallest// prime factor of '8' and '16'// is '2' so we put SPF[8] = 2 ,// SPF[16] = 2 ]$isprime = array_fill(0, $MAX_SIZE, true);$prime = array();$SPF = array_fill(0, $MAX_SIZE, 0); // function generate all prime number// less then N in O(n)function manipulated_seive($N){    global $isprime, $MAX_SIZE,                     $SPF, $prime;         // 0 and 1 are not prime    $isprime[0] = $isprime[1] = false;     // Fill rest of the entries    for ($i = 2; $i < $N; $i++)    {             // If isPrime[i] == True then        // i is prime number        if ($isprime[$i])        {                     // put i into prime[] vector            array_push($prime, $i);             // A prime number is its own             // smallest prime factor            $SPF[$i] = $i;        }                 // Remove all multiples of i*prime[j]        // which are not prime by making is        // Prime[i * prime[j]] = false and put        // smallest prime factor of i*Prime[j]        // as prime[j] [ for exp :let i = 5 , j = 0 ,        // prime[j] = 2 [ i*prime[j] = 10 ]        // so smallest prime factor of '10' is '2'        // that is prime[j] ] this loop run only         // one time for number which are not prime        $j = 0;        while ($j < count($prime) &&               $i * $prime[$j] < $N &&               $prime[$j] <= $SPF[$i])       {            $isprime[$i * $prime[$j]] = false;             // put smallest prime factor of i*prime[j]            $SPF[$i * $prime[$j]] = $prime[$j];                         $j += 1;        }    }}         // Driver Code$N = 13; // Must be less than MAX_SIZE manipulated_seive($N); // print all prime number less then N$i = 0;while ($i < count($prime) &&       $prime[$i] <= $N){    print($prime[$i] . " ");    $i += 1;}     // This code is contributed by mits?>
<script>   // Javascript program to generate all  // prime numbers smaller than N in O(N)   const MAX_SIZE = 1000001;   // isPrime[] : isPrime[i] is true if the number is prime  // prime[] : stores all prime numbers less than N  // SPF[] that store smallest prime factor of number  // [for Exp : smallest prime factor of '8' and '16'  // is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]  var isPrime = Array.from({ length: MAX_SIZE }, (_, i) => true);  var prime = [];  var SPF = Array.from({ length: MAX_SIZE });   // function that generates all prime number  // less than N in O(N)  function manipulated_sieve(N) {       // 0 and 1 are not prime    isPrime[0] = isPrime[1] = true;     // Fill rest of the entries    for (let i = 2; i < N; i++)    {      // If isPrime[i] === true,      // then i is a prime number      if (isPrime[i])      {        // put i into prime[] array        prime.push(i);         // A prime number is its own smallest        // prime factor        SPF[i] = i;      }       // Remove all multiples of  i*prime[j] which are      // not prime by making isPrime[i*prime[j]] = false      // and put smallest prime factor of i*Prime[j] as prime[j]      // [ for exp :let  i = 5 , j = 0 , prime[j] = 2 [ i*prime[j] = 10 ]      // so smallest prime factor of '10' is '2' that is prime[j] ]      // this loop run only one time for number which are not prime      for (        let j = 0;        j < prime.length && i * prime[j] < N && prime[j] <= SPF[i];        j++      ) {        isPrime[i * prime[j]] = false;         // put smallest prime factor of i*prime[j]        SPF[i * prime[j]] = prime[j];      }    }  }   // Driver Code  var N = 13; // Must be less than MAX_SIZE   manipulated_sieve(N);     // print all prime numbers less than N  for (let i = 0; i < prime.length && prime[i] <= N; i++) {    document.write(prime[i] + " ");  }</script>
Output :
2 3 5 7 11
Illustration:
isPrime[0] = isPrime[1] = 0

After i = 2 iteration :
isPrime[]   [F, F, T, T, F, T, T, T] 
SPF[]       [0, 0, 2, 0, 2, 0, 0, 0]
     index   0  1  2  3  4  5  6  7

After i = 3 iteration :
isPrime[]  [F, F, T, T, F, T, F, T, T, F ]
SPF[]      [0, 0, 2, 3, 2, 0, 2, 0, 0, 3 ]
  index     0  1  2  3  4  5  6  7  8  9

After i = 4 iteration :
isPrime[]  [F, F, T, T, F, T, F, T, F, F]
SPF[]      [0, 0, 2, 3, 2, 0, 2, 0, 2, 3]
  index     0  1  2  3  4  5  6  7  8  9
This article is contributed by Divyanshu Srivastava and Nishant Singh. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
Prashuk156
rituraj_jain
Mithun Kumar
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