Eratosthenes算法的经典Sieve需要O(N log(log N))时间才能找到所有小于N的素数。在本文中，我们讨论了一种在O(N)时间工作的改进的Sieve。
例子 ：
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
Eratosthenes算法的处理过的筛网的工作方式如下：
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 j*p as non_prime.
Mark smallest prime factor of j*p as j
以下是上述想法的实现。
C++
// C++ program to generate all prime numbers
// less than N in O(N)
#include
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 ]
vectorisprime(MAX_SIZE , true);
vectorprime;
vectorSPF(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 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 Vectorisprime = new Vector<>(MAX_SIZE);
static Vectorprime = new Vector<>();
static VectorSPF = 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 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 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 Code
if __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 JainPHP
输出 ：
2 3 5 7 11
插图：
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