reporting all prime numbers less than n

Question

I need to print all the prime numbers less than a given number n. I can use sieve of Eratothenes but the running time of that algorithm IS NOT O(n). Is there any O(n) time s

Answer 1

The Sieve of Eratosthenes has time complexity O(n log log n). The function log log n grows very slowly; for example, log(log(10^9)) = 3. That means you can effectively treat the log log n part of the time complexity as a constant, and ignore it, giving a time complexity of "nearly" O(n).

There are various algorithms that operate in time O(n) or O(n / log log n), including Pritchard's wheel sieves and Atkins' sieve. But constant factors generally make those algorithms slower in practice than the Sieve of Eratosthenes. Unless you need extreme speed, and you know what you are doing, and you are willing to spend lots of time doing it, the practical answer is the Sieve of Eratosthenes.

Your question says that you are going to print the list of primes. In that case, output will dominate the run time of any algorithm you choose. So do yourself a favor and implement a simple Sieve of Eratosthenes.