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++
// 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 than 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 than N for (int i=0; i<prime.size() && prime[i] <= N ; i++) cout << prime[i] << " "; return 0;} |
Java
// 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 than 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 than N for (int i=0; i<prime.size() && prime.get(i) <= N ; i++) System.out.print(prime.get(i) + " "); }} |
Python3
# 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_SIZE prime = [] SPF = [None] * (MAX_SIZE) # function generate all prime number # less than 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 than N i = 0 while i < len(prime) and prime[i] <= N: print(prime[i], end = " ") i += 1 # This code is contributed by Rituraj Jain |
C#
// C# program to generate all prime numbers// less than N in O(N)using System;using System.Collections.Generic;class Test { static 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 List<bool> isprime = new List<bool>(MAX_SIZE); static List<int> prime = new List<int>(); static List<int> SPF = new List<int>(MAX_SIZE); // method generate all prime number less than N in O(n) static void manipulated_seive(int N) { // 0 and 1 are not prime isprime[0] = false; isprime[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[i]) { // put i into prime[] vector prime.Add(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 (int j = 0; j < prime.Count && 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 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 than N for (int i = 0; i < prime.Count && prime[i] <= N; i++) Console.Write(prime[i] + " "); }}// This code is contributed by phasing17 |
PHP
<?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, $MAX_SIZE, 0); // function generate all prime number // less than N in O(n) function manipulated_seive($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 than N $i = 0;while ($i < count($prime) && $prime[$i] <= $N){ print($prime[$i] . " "); $i += 1;} // This code is contributed by mits?> |
Javascript
<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
Auxiliary Space: O(1)
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
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