这是我能想到的最好的算法。
def get_primes(n):
numbers = set(range(n, 1, -1))
primes = []
while numbers:
p = numbers.pop()
primes.append(p)
numbers.difference_update(set(range(p*2, n+1, p)))
return primes
>>> timeit.Timer(stmt='get_primes.get_primes(1000000)', setup='import get_primes').timeit(1)
1.1499958793645562
还能做得更快吗?
这段代码有一个缺陷:由于numbers是一个无序集,不能保证numbers.pop()将从集合中移除最低的数字。尽管如此,它还是适用于(至少对我来说)一些输入数字:
>>> sum(get_primes(2000000))
142913828922L
#That's the correct sum of all numbers below 2 million
>>> 529 in get_primes(1000)
False
>>> 529 in get_primes(530)
True
使用Numpy实现的半筛子略有不同:
http://rebrained.com/?p=458
import math
import numpy
def prime6(upto):
primes=numpy.arange(3,upto+1,2)
isprime=numpy.ones((upto-1)/2,dtype=bool)
for factor in primes[:int(math.sqrt(upto))]:
if isprime[(factor-2)/2]: isprime[(factor*3-2)/2:(upto-1)/2:factor]=0
return numpy.insert(primes[isprime],0,2)
有人能把这个和其他时间比较一下吗?在我的机器上,它似乎与其他Numpy半筛相当。
使用Sundaram的Sieve,我想我打破了pure-Python的记录:
def sundaram3(max_n):
numbers = range(3, max_n+1, 2)
half = (max_n)//2
initial = 4
for step in xrange(3, max_n+1, 2):
for i in xrange(initial, half, step):
numbers[i-1] = 0
initial += 2*(step+1)
if initial > half:
return [2] + filter(None, numbers)
Comparasion:
C:\USERS>python -m timeit -n10 -s "import get_primes" "get_primes.get_primes_erat(1000000)"
10 loops, best of 3: 710 msec per loop
C:\USERS>python -m timeit -n10 -s "import get_primes" "get_primes.daniel_sieve_2(1000000)"
10 loops, best of 3: 435 msec per loop
C:\USERS>python -m timeit -n10 -s "import get_primes" "get_primes.sundaram3(1000000)"
10 loops, best of 3: 327 msec per loop
下面是Eratosthenes的一个numpy版本,具有良好的复杂度(低于排序长度为n的数组)和向量化。与@unutbu相比,用46微秒就可以找到100万以下的所有质数。
import numpy as np
def generate_primes(n):
is_prime = np.ones(n+1,dtype=bool)
is_prime[0:2] = False
for i in range(int(n**0.5)+1):
if is_prime[i]:
is_prime[i**2::i]=False
return np.where(is_prime)[0]
计时:
import time
for i in range(2,10):
timer =time.time()
generate_primes(10**i)
print('n = 10^',i,' time =', round(time.time()-timer,6))
>> n = 10^ 2 time = 5.6e-05
>> n = 10^ 3 time = 6.4e-05
>> n = 10^ 4 time = 0.000114
>> n = 10^ 5 time = 0.000593
>> n = 10^ 6 time = 0.00467
>> n = 10^ 7 time = 0.177758
>> n = 10^ 8 time = 1.701312
>> n = 10^ 9 time = 19.322478