Efficient Sieve of Eratosthenes

//https://www.geeksforgeeks.org/sieve-eratosthenes-0n-time-complexity/
#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 function
int main()
{
    int N = 13 ; // Must be less than MAX_SIZE

    manipulated_seive(N);

    // pint all prime number less then N
    for (int i=0; i<prime.size() && prime[i] <= N ; i++)
        cout << prime[i] << " ";

    return 0;
}