Sieve Of Eratosthenes

The sieve of Eratosthenes is an algorithm for finding all prime numbers up to any given limit.

Algorithm

Start from the first prime 2, mark all the multiples of 2 as composite (i.e. not prime). So we mark 2*2, 2*3, 2*4, ... until we reach the greater number in consideration.

Then the next first number after 2 that is not marked as composite is a prime. It's 3. Then we do the same thing, marking all the multiples of 3 as composite, 3*2, 3*3, 3*4, .... Note that since 2*3 has already been marked, so we shouldn't mark it again. Thus, instead of start marking from 2 * 3, we should start from 3 * 3, i.e. sqrt(3).

Then the next prime is 5, we mark all its multiples as composite. Again, we should start marking from sqrt(5) since 2*5, 3*5, 4*5 have all been marked already.

And so on. Until we've visited all the numbers within the maximum number.

Implementation

// OJ: https://leetcode.com/problems/count-primes/
// Author: github.com/lzl124631x
// Time: O(NloglogN)
// Space: O(N)
// Ref: https://leetcode.com/problems/count-primes/solution/
class Solution {
public:
    int countPrimes(int n) {
        if (n <= 1) return 0;
        vector<bool> prime(n, true);
        int ans = 0, bound = sqrt(n);
        for (int i = 2; i < n; ++i) {
            if (!prime[i]) continue;
            ++ans;
            if (i > bound) continue; // If i > bound, then i * i must be greater than `n`, skip. This can prevent overflow caused by `i * i`.
            for (int j = i * i; j < n; j += i) prime[j] = false; // note that we start from `i * i` instead of `2` because all multiples of `2, 3, ..., (i-1)` must be marked already
        }
        return ans;
    }
};

Problems

Reference

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