Cảnh báo: timeit kết quả có thể thay đổi do sự khác biệt về phần cứng hoặc phiên bản của Python.
Dưới đây là một kịch bản so sánh một số triển khai:
Rất cám ơn stephan vì đã chú ý đến sieve_wheel_30. Tín dụng dành cho Robert William Hanks cho primesfrom2to, primesfrom3to, rwh_primes, rwh_primes1 và rwh_primes2.
Trong số các phương thức Python đơn giản được thử nghiệm, với psyco , với n = 1000000,
rwh_primes1 là thử nghiệm nhanh nhất.
+---------------------+-------+
| Method | ms |
+---------------------+-------+
| rwh_primes1 | 43.0 |
| sieveOfAtkin | 46.4 |
| rwh_primes | 57.4 |
| sieve_wheel_30 | 63.0 |
| rwh_primes2 | 67.8 |
| sieveOfEratosthenes | 147.0 |
| ambi_sieve_plain | 152.0 |
| sundaram3 | 194.0 |
+---------------------+-------+
Trong số các phương thức Python đơn giản được thử nghiệm, không có psyco , với n = 1000000,
rwh_primes2 là nhanh nhất.
+---------------------+-------+
| Method | ms |
+---------------------+-------+
| rwh_primes2 | 68.1 |
| rwh_primes1 | 93.7 |
| rwh_primes | 94.6 |
| sieve_wheel_30 | 97.4 |
| sieveOfEratosthenes | 178.0 |
| ambi_sieve_plain | 286.0 |
| sieveOfAtkin | 314.0 |
| sundaram3 | 416.0 |
+---------------------+-------+
Trong tất cả các phương pháp được thử nghiệm, cho phép numpy , với n = 1000000,
primesfrom2to là thử nghiệm nhanh nhất.
+---------------------+-------+
| Method | ms |
+---------------------+-------+
| primesfrom2to | 15.9 |
| primesfrom3to | 18.4 |
| ambi_sieve | 29.3 |
+---------------------+-------+
Thời gian được đo bằng lệnh:
python -mtimeit -s"import primes" "primes.{method}(1000000)"
với {method}thay thế bởi mỗi tên phương thức.
primes.py:
#!/usr/bin/env python
import psyco; psyco.full()
from math import sqrt, ceil
import numpy as np
def rwh_primes(n):
# /programming/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
""" Returns a list of primes < n """
sieve = [True] * n
for i in xrange(3,int(n**0.5)+1,2):
if sieve[i]:
sieve[i*i::2*i]=[False]*((n-i*i-1)/(2*i)+1)
return [2] + [i for i in xrange(3,n,2) if sieve[i]]
def rwh_primes1(n):
# /programming/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
""" Returns a list of primes < n """
sieve = [True] * (n/2)
for i in xrange(3,int(n**0.5)+1,2):
if sieve[i/2]:
sieve[i*i/2::i] = [False] * ((n-i*i-1)/(2*i)+1)
return [2] + [2*i+1 for i in xrange(1,n/2) if sieve[i]]
def rwh_primes2(n):
# /programming/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
""" Input n>=6, Returns a list of primes, 2 <= p < n """
correction = (n%6>1)
n = {0:n,1:n-1,2:n+4,3:n+3,4:n+2,5:n+1}[n%6]
sieve = [True] * (n/3)
sieve[0] = False
for i in xrange(int(n**0.5)/3+1):
if sieve[i]:
k=3*i+1|1
sieve[ ((k*k)/3) ::2*k]=[False]*((n/6-(k*k)/6-1)/k+1)
sieve[(k*k+4*k-2*k*(i&1))/3::2*k]=[False]*((n/6-(k*k+4*k-2*k*(i&1))/6-1)/k+1)
return [2,3] + [3*i+1|1 for i in xrange(1,n/3-correction) if sieve[i]]
def sieve_wheel_30(N):
# http://zerovolt.com/?p=88
''' Returns a list of primes <= N using wheel criterion 2*3*5 = 30
Copyright 2009 by zerovolt.com
This code is free for non-commercial purposes, in which case you can just leave this comment as a credit for my work.
If you need this code for commercial purposes, please contact me by sending an email to: info [at] zerovolt [dot] com.'''
__smallp = ( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,
149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,
229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311,
313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401,
409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491,
499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,
601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683,
691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,
809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997)
wheel = (2, 3, 5)
const = 30
if N < 2:
return []
if N <= const:
pos = 0
while __smallp[pos] <= N:
pos += 1
return list(__smallp[:pos])
# make the offsets list
offsets = (7, 11, 13, 17, 19, 23, 29, 1)
# prepare the list
p = [2, 3, 5]
dim = 2 + N // const
tk1 = [True] * dim
tk7 = [True] * dim
tk11 = [True] * dim
tk13 = [True] * dim
tk17 = [True] * dim
tk19 = [True] * dim
tk23 = [True] * dim
tk29 = [True] * dim
tk1[0] = False
# help dictionary d
# d[a , b] = c ==> if I want to find the smallest useful multiple of (30*pos)+a
# on tkc, then I need the index given by the product of [(30*pos)+a][(30*pos)+b]
# in general. If b < a, I need [(30*pos)+a][(30*(pos+1))+b]
d = {}
for x in offsets:
for y in offsets:
res = (x*y) % const
if res in offsets:
d[(x, res)] = y
# another help dictionary: gives tkx calling tmptk[x]
tmptk = {1:tk1, 7:tk7, 11:tk11, 13:tk13, 17:tk17, 19:tk19, 23:tk23, 29:tk29}
pos, prime, lastadded, stop = 0, 0, 0, int(ceil(sqrt(N)))
# inner functions definition
def del_mult(tk, start, step):
for k in xrange(start, len(tk), step):
tk[k] = False
# end of inner functions definition
cpos = const * pos
while prime < stop:
# 30k + 7
if tk7[pos]:
prime = cpos + 7
p.append(prime)
lastadded = 7
for off in offsets:
tmp = d[(7, off)]
start = (pos + prime) if off == 7 else (prime * (const * (pos + 1 if tmp < 7 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# 30k + 11
if tk11[pos]:
prime = cpos + 11
p.append(prime)
lastadded = 11
for off in offsets:
tmp = d[(11, off)]
start = (pos + prime) if off == 11 else (prime * (const * (pos + 1 if tmp < 11 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# 30k + 13
if tk13[pos]:
prime = cpos + 13
p.append(prime)
lastadded = 13
for off in offsets:
tmp = d[(13, off)]
start = (pos + prime) if off == 13 else (prime * (const * (pos + 1 if tmp < 13 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# 30k + 17
if tk17[pos]:
prime = cpos + 17
p.append(prime)
lastadded = 17
for off in offsets:
tmp = d[(17, off)]
start = (pos + prime) if off == 17 else (prime * (const * (pos + 1 if tmp < 17 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# 30k + 19
if tk19[pos]:
prime = cpos + 19
p.append(prime)
lastadded = 19
for off in offsets:
tmp = d[(19, off)]
start = (pos + prime) if off == 19 else (prime * (const * (pos + 1 if tmp < 19 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# 30k + 23
if tk23[pos]:
prime = cpos + 23
p.append(prime)
lastadded = 23
for off in offsets:
tmp = d[(23, off)]
start = (pos + prime) if off == 23 else (prime * (const * (pos + 1 if tmp < 23 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# 30k + 29
if tk29[pos]:
prime = cpos + 29
p.append(prime)
lastadded = 29
for off in offsets:
tmp = d[(29, off)]
start = (pos + prime) if off == 29 else (prime * (const * (pos + 1 if tmp < 29 else 0) + tmp) )//const
del_mult(tmptk[off], start, prime)
# now we go back to top tk1, so we need to increase pos by 1
pos += 1
cpos = const * pos
# 30k + 1
if tk1[pos]:
prime = cpos + 1
p.append(prime)
lastadded = 1
for off in offsets:
tmp = d[(1, off)]
start = (pos + prime) if off == 1 else (prime * (const * pos + tmp) )//const
del_mult(tmptk[off], start, prime)
# time to add remaining primes
# if lastadded == 1, remove last element and start adding them from tk1
# this way we don't need an "if" within the last while
if lastadded == 1:
p.pop()
# now complete for every other possible prime
while pos < len(tk1):
cpos = const * pos
if tk1[pos]: p.append(cpos + 1)
if tk7[pos]: p.append(cpos + 7)
if tk11[pos]: p.append(cpos + 11)
if tk13[pos]: p.append(cpos + 13)
if tk17[pos]: p.append(cpos + 17)
if tk19[pos]: p.append(cpos + 19)
if tk23[pos]: p.append(cpos + 23)
if tk29[pos]: p.append(cpos + 29)
pos += 1
# remove exceeding if present
pos = len(p) - 1
while p[pos] > N:
pos -= 1
if pos < len(p) - 1:
del p[pos+1:]
# return p list
return p
def sieveOfEratosthenes(n):
"""sieveOfEratosthenes(n): return the list of the primes < n."""
# Code from: <dickinsm@gmail.com>, Nov 30 2006
# http://groups.google.com/group/comp.lang.python/msg/f1f10ced88c68c2d
if n <= 2:
return []
sieve = range(3, n, 2)
top = len(sieve)
for si in sieve:
if si:
bottom = (si*si - 3) // 2
if bottom >= top:
break
sieve[bottom::si] = [0] * -((bottom - top) // si)
return [2] + [el for el in sieve if el]
def sieveOfAtkin(end):
"""sieveOfAtkin(end): return a list of all the prime numbers <end
using the Sieve of Atkin."""
# Code by Steve Krenzel, <Sgk284@gmail.com>, improved
# Code: https://web.archive.org/web/20080324064651/http://krenzel.info/?p=83
# Info: http://en.wikipedia.org/wiki/Sieve_of_Atkin
assert end > 0
lng = ((end-1) // 2)
sieve = [False] * (lng + 1)
x_max, x2, xd = int(sqrt((end-1)/4.0)), 0, 4
for xd in xrange(4, 8*x_max + 2, 8):
x2 += xd
y_max = int(sqrt(end-x2))
n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1
if not (n & 1):
n -= n_diff
n_diff -= 2
for d in xrange((n_diff - 1) << 1, -1, -8):
m = n % 12
if m == 1 or m == 5:
m = n >> 1
sieve[m] = not sieve[m]
n -= d
x_max, x2, xd = int(sqrt((end-1) / 3.0)), 0, 3
for xd in xrange(3, 6 * x_max + 2, 6):
x2 += xd
y_max = int(sqrt(end-x2))
n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1
if not(n & 1):
n -= n_diff
n_diff -= 2
for d in xrange((n_diff - 1) << 1, -1, -8):
if n % 12 == 7:
m = n >> 1
sieve[m] = not sieve[m]
n -= d
x_max, y_min, x2, xd = int((2 + sqrt(4-8*(1-end)))/4), -1, 0, 3
for x in xrange(1, x_max + 1):
x2 += xd
xd += 6
if x2 >= end: y_min = (((int(ceil(sqrt(x2 - end))) - 1) << 1) - 2) << 1
n, n_diff = ((x*x + x) << 1) - 1, (((x-1) << 1) - 2) << 1
for d in xrange(n_diff, y_min, -8):
if n % 12 == 11:
m = n >> 1
sieve[m] = not sieve[m]
n += d
primes = [2, 3]
if end <= 3:
return primes[:max(0,end-2)]
for n in xrange(5 >> 1, (int(sqrt(end))+1) >> 1):
if sieve[n]:
primes.append((n << 1) + 1)
aux = (n << 1) + 1
aux *= aux
for k in xrange(aux, end, 2 * aux):
sieve[k >> 1] = False
s = int(sqrt(end)) + 1
if s % 2 == 0:
s += 1
primes.extend([i for i in xrange(s, end, 2) if sieve[i >> 1]])
return primes
def ambi_sieve_plain(n):
s = range(3, n, 2)
for m in xrange(3, int(n**0.5)+1, 2):
if s[(m-3)/2]:
for t in xrange((m*m-3)/2,(n>>1)-1,m):
s[t]=0
return [2]+[t for t in s if t>0]
def sundaram3(max_n):
# /programming/2068372/fastest-way-to-list-all-primes-below-n-in-python/2073279#2073279
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)
################################################################################
# Using Numpy:
def ambi_sieve(n):
# http://tommih.blogspot.com/2009/04/fast-prime-number-generator.html
s = np.arange(3, n, 2)
for m in xrange(3, int(n ** 0.5)+1, 2):
if s[(m-3)/2]:
s[(m*m-3)/2::m]=0
return np.r_[2, s[s>0]]
def primesfrom3to(n):
# /programming/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
""" Returns a array of primes, p < n """
assert n>=2
sieve = np.ones(n/2, dtype=np.bool)
for i in xrange(3,int(n**0.5)+1,2):
if sieve[i/2]:
sieve[i*i/2::i] = False
return np.r_[2, 2*np.nonzero(sieve)[0][1::]+1]
def primesfrom2to(n):
# /programming/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
""" Input n>=6, Returns a array of primes, 2 <= p < n """
sieve = np.ones(n/3 + (n%6==2), dtype=np.bool)
sieve[0] = False
for i in xrange(int(n**0.5)/3+1):
if sieve[i]:
k=3*i+1|1
sieve[ ((k*k)/3) ::2*k] = False
sieve[(k*k+4*k-2*k*(i&1))/3::2*k] = False
return np.r_[2,3,((3*np.nonzero(sieve)[0]+1)|1)]
if __name__=='__main__':
import itertools
import sys
def test(f1,f2,num):
print('Testing {f1} and {f2} return same results'.format(
f1=f1.func_name,
f2=f2.func_name))
if not all([a==b for a,b in itertools.izip_longest(f1(num),f2(num))]):
sys.exit("Error: %s(%s) != %s(%s)"%(f1.func_name,num,f2.func_name,num))
n=1000000
test(sieveOfAtkin,sieveOfEratosthenes,n)
test(sieveOfAtkin,ambi_sieve,n)
test(sieveOfAtkin,ambi_sieve_plain,n)
test(sieveOfAtkin,sundaram3,n)
test(sieveOfAtkin,sieve_wheel_30,n)
test(sieveOfAtkin,primesfrom3to,n)
test(sieveOfAtkin,primesfrom2to,n)
test(sieveOfAtkin,rwh_primes,n)
test(sieveOfAtkin,rwh_primes1,n)
test(sieveOfAtkin,rwh_primes2,n)
Chạy các bài kiểm tra kịch bản mà tất cả các triển khai đều cho kết quả như nhau.
