Count the number of prime numbers less than a non-negative number, n.

思路

  1. Let’s start with a isPrime function. To determine if a number is prime, we need to check if it is not divisible by any number less than n. The runtime complexity of isPrime function would be O(n) and hence counting the total prime numbers up to n would be O(n2). Could we do better?

  2. As we know the number must not be divisible by any number > n / 2, we can immediately cut the total iterations half by dividing only up to n / 2. Could we still do better?

  3. Let’s write down all of 12’s factors:

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    2 × 6 = 12
    3 × 4 = 12
    4 × 3 = 12
    6 × 2 = 12

    As you can see, calculations of 4 × 3 and 6 × 2 are not necessary. Therefore, we only need to consider factors up to √n because, if n is divisible by some number p, then n = p × q and since p ≤ q, we could derive that p ≤ √n.

    Our total runtime has now improved to O(n1.5), which is slightly better. Is there a faster approach?

  1. The Sieve of Eratosthenes is one of the most efficient ways to find all prime numbers up to n. But don’t let that name scare you, I promise that the concept is surprisingly simple.

    img

    Sieve of Eratosthenes: algorithm steps for primes below 121. “Sieve of Eratosthenes Animation“ by SKopp is licensed under CC BY 2.0.

    We start off with a table of n numbers. Let’s look at the first number, 2. We know all multiples of 2 must not be primes, so we mark them off as non-primes. Then we look at the next number, 3. Similarly, all multiples of 3 such as 3 × 2 = 6, 3 × 3 = 9, … must not be primes, so we mark them off as well. Now we look at the next number, 4, which was already marked off. What does this tell you? Should you mark off all multiples of 4 as well?

  2. 4 is not a prime because it is divisible by 2, which means all multiples of 4 must also be divisible by 2 and were already marked off. So we can skip 4 immediately and go to the next number, 5. Now, all multiples of 5 such as 5 × 2 = 10, 5 × 3 = 15, 5 × 4 = 20, 5 × 5 = 25, … can be marked off. There is a slight optimization here, we do not need to start from 5 × 2 = 10. Where should we start marking off?

  3. In fact, we can mark off multiples of 5 starting at 5 × 5 = 25, because 5 × 2 = 10 was already marked off by multiple of 2, similarly 5 × 3 = 15 was already marked off by multiple of 3. Therefore, if the current number is p, we can always mark off multiples of p starting at p2, then in increments of p: p2 + p, p2 + 2p, … Now what should be the terminating loop condition?

  4. It is easy to say that the terminating loop condition is p < n, which is certainly correct but not efficient. Do you still remember Hint #3?

这也就是官方给出的思路，所以我先试了一下，给出 p ≤ √n的情况，能不能通过。

O(n^1.5), Time limit exceeded. 

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import math
class Solution(object):
    def countPrimes(self, n):
        """
        :type n: int
        :rtype: int
        """
        count = 0
        
        for i in range(2,n):
            flag = 1
            k = 2
            while (k * k <= i):
                if (i % k == 0):
                    flag = 0
                    break
                k += 1
            if (flag):
                count += 1
        return count

答案显然是不行。

接下来我们试一下图中所给的方法。

要注意，primes数组存储的应当是布尔量而不是普通的整数。如果是普通的整数，会造成memory limit exceed的错误。

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class Solution(object):
    def countPrimes(self, n):
        """
        :type n: int
        :rtype: int
        """
        count = 0
        primes = []
        for i in range(n+1):
            primes.append(True)
        
        i = 2
        
        while (i * i < n):
            if (primes[i] == True):
                k = i
                while(i * k <= n):
                    if (primes[i * k] > 0):
                        primes[i * k] = False
                    k += 1
            i += 1
        for i in range(2,n):
            if (primes[i] == True):
                count += 1
                
        return count