Python Cookbook 2Nd Edition Jun 1002005 [Electronic resources] نسخه متنی

Recipe 18.10. Computing Prime Numbers

Credit: David Eppstein, Tim Peters, Alex Martelli, Wim
Stolker, Kazuo Moriwaka, Hallvard Furuseth, Pierre Denis, Tobias
Klausmann, David Lees, Raymond Hettinger

Problem

You need to compute an unbounded
sequence of all primes, or the list of all primes that are less than
a certain threshold.

Solution

To compute an unbounded sequence, a generator is the natural Pythonic
approach, and the Sieve of Eratosthenes, using a dictionary as the
supporting data structure, is the natural algorithm to use:

import itertools
def eratosthenes( ):
'''Yields the sequence of prime numbers via the Sieve of Eratosthenes.'''
D = {  }  # map each composite integer to its first-found prime factor
for q in itertools.count(2):     # q gets 2, 3, 4, 5, ... ad infinitum
p = D.pop(q, None)
if p is None:
# q not a key in D, so q is prime, therefore, yield it
yield q
# mark q squared as not-prime (with q as first-found prime factor)
D[q*q] = q
else:
# let x <- smallest (N*p)+q which wasn't yet known to be composite
# we just learned x is composite, with p first-found prime factor,
# since p is the first-found prime factor of q -- find and mark it
x = p + q
while x in D:
x += p
D[x] = p

Discussion

To compute all primes up to a predefined threshold, rather than an
unbounded sequence, it's reasonable to wonder if
it's possible to use a faster way than good old
Eratosthenes, even in the smart variant shown as the
"Solution". Here is a function that
uses a few speed-favoring tricks, such as a hard-coded tuple of the
first few primes:

def primes_less_than(N):
# make `primes' a list of known primes < N
primes = [x for x in (2, 3, 5, 7, 11, 13) if x < N]
if N <= 17: return primes
# candidate primes are all odd numbers less than N and over 15,
# not divisible by the first few known primes, in descending order
candidates = [x for x in xrange((N-2)|1, 15, -2)
if x % 3 and x % 5 and x % 7 and x % 11 and x % 13]
# make `top' the biggest number that we must check for compositeness
top = int(N ** 0.5)
while (top+1)*(top+1) <= N:
top += 1
# main loop, weeding out non-primes among the remaining candidates
while True:
# get the smallest candidate: it must be a prime
p = candidates.pop( )
primes.append(p)
if p > top:
break
# remove all candidates which are divisible by the newfound prime
candidates = filter(p._ _rmod_ _, candidates)
# all remaining candidates are prime, add them (in ascending order)
candidates.reverse( )
primes.extend(candidates)
return primes

On a typical small task such as looping over all primes up to 8,192,
eratosthenes (on an oldish 1.2 GHz Athlon PC, with
Python 2.4) takes 22 milliseconds, while
primes_less_than takes 9.7; so, the slight
trickery and limitations of primes_less_than can
pay for themselves quite handsomely if generating such primes is a
bottleneck in your program. Be aware, however, that
eratosthenes scales better. If you need all primes
up to 199,999, eratosthenes will deliver them in
0.88 seconds, while primes_less_than takes 0.65.

Since primes_less_than's little
speed-up tricks can help, it's natural to wonder
whether a perhaps simpler trick can be retrofitted into
eratosthenes as well. For example, we might simply
avoid wasting work on a lot of even numbers,
concentrating on odd numbers only, beyond the initial
2. In other words:

def erat2( ):
D = {  }
yield 2
for q in itertools.islice(itertools.count(3), 0, None, 2):
p = D.pop(q, None)
if p is None:
D[q*q] = q
yield q
else:
x = p + q
while x in D or not (x&1):
x += p
D[x] = p

And indeed, erat2 takes 16 milliseconds, versus
eratosthenes' 22, to get primes
up to 8,192; 0.49 seconds, versus
eratosthenes' 0.88, to get primes
up to 199,999. In other words, erat2 scales just
as well as eratosthenes and is always
approximately 25% faster. Incidentally, if you're
wondering whether it might be even faster to program at a slightly
lower level, with q = 3 to start, a while
True as the loop header, and a q += 2 at
the end of the loop, don't worrythe slightly
higher-level approach using
itertools'
count and islice functions is
repeatedly approximately 4% faster. Other languages may impose a
performance penalty for programming with higher abstraction, Python
rewards you for doing that.

You might keep pushing the same idea yet further, avoiding multiples
of 3 as well as even numbers, and so on. However,
it would be an exercise in diminishing returns: greater and greater
complication for smaller and smaller gain. It's
better to quit while we're ahead!

If you're into one liners, you might want to study
the following:

def primes_oneliner(N):
aux = {  }
return [aux.setdefault(p, p) for p in range(2, N)
if 0 not in [p%d for d in aux if p>=d+d]]

Be aware that one liners, even clever ones, are generally anything
but speed demons! primes_oneliner takes 2.9 seconds
to complete the same small task (computing primes up to 8,192) which,
eratosthenes does in 22 milliseconds, and
primes_less_than in 9.7so,
you're slowing things down by 130 to 300 times just
for the fun of using a clever, opaque one liner, which is clearly not
a sensible tradeoff. Clever one liners can be instructive but should
almost never be used in production code, not just because
they're terse and make maintenance harder than
straightforward coding (which is by far the main reason), but also
because of the speed penalties they may entail.

While prime numbers, and number theory more generally, used to be
considered purely theoretical problems, nowadays they have plenty of
practical applications, starting with cryptography.

See Also

To explore both number theory and its applications, the best book is
probably Kenneth Rosen, Elementary Number Theory and Its
Applications (Addison-Wesley); http://www.utm.edu/research/primes/ for more
information about prime numbers.