Implement Sieve of Eratosthenes to Generate Prime Numbers in C++

The Sieve of Eratosthenes algorithm is one of the most efficient ways to find prime numbers smaller than n when n is smaller than around 10 million. It follows a simple process of marking the multiples of already prime numbers as false i.e. non-prime numbers.

In this article, we have a given number as 'num'. Our task is to find all the prime numbers less than or equal to num using the Sieve of Eratosthenes algorithm in C++.

Example

Here is an example to find prime numbers less than 10:

Input:
num = 10

Output:
2 3 5 7

The explanation of the above example is as follows:

Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

i = 2 = Prime number
After removing all multiples of 2:
2, 3, 5, 7, 9

i = 3 = Prime number
After removing all multiples of 3:
2, 3, 5, 7

Prime numbers: 2, 3, 5, 7

Steps to Implement Sieve of Eratosthenes

Here are the steps to implement the Sieve of Eratosthenes algorithm to find the the prime numbers:

• We have defined a prime() function that accepts the given number num as argument.
• We have initialized the first two elements as false i.e. '0' and '1' are not prime numbers.
• Then we check if the current element 'i' is prime or not. If it is prime, then we mark their multiples as not prime.
• Then we have returned all the prime numbers using the second for loop.

C++ Implementation of Sieve of Eratosthenes

Here is the C++ code implementing the above steps to implement Sieve of Eratosthenes algorithm.

#include <bits/stdc++.h>
using namespace std;

void prime(int num) {
    vector<bool> pno(num + 1, true);
    pno[0] = pno[1] = false;

    for (int i = 2; i * i <= num; i++) {
        if (pno[i]) {
            for (int j = i * i; j <= num; j += i)
                pno[j] = false;
        }
    }

    for (int i = 2; i <= num; i++) {
        if (pno[i])
            cout << i << " ";
    }
}

int main() {
    int num = 40;
    cout << "The prime numbers upto " << num << " are: ";
    prime(num);
    return 0;
}

The output of the above code is:

The prime numbers upto 30 are: 2 3 5 7 11 13 17 19 23 29 31 37 

Complexity of Sieve of Eratosthenes Algorithm

• Time Complexity: The time complexity of the Sieve of Eratosthenes algorithm is O(n log(logn)).
• Space Complexity: The space complexity of the Sieve of Eratosthenes algorithm is O(n).