Sieve of Eratosthenes - Finding Primes Python

Question

Just to clarify  this is not a homework problem     I wanted to find primes for a math application I am building  amp  came across Sieve of Eratosthenes approach    I have written an implementation of it in Python  But it s terribly slow  For say  if I want to find all primes less than 2 million  It takes   20 mins   I stopped it at this point   How can I speed this up   def primes sieve limit       limitn   limit 1     primes   range 2  limitn       for i in primes          factors   range i  limitn  i          for f in factors 1                if f in primes                  primes remove f      return primes  print primes sieve 2000    UPDATE  I ended up doing profiling on this code  amp  found that quite a lot of time was spent on removing an element from the list  Quite understandable considering it has to traverse the entire list  worst-case  to find the element  amp  then remove it and then readjust the list  maybe some copy goes on    Anyway  I chucked out list for dictionary  My new implementation -   def primes sieve1 limit       limitn   limit 1     primes   dict       for i in range 2  limitn   primes i    True      for i in primes          factors   range i limitn  i          for f in factors 1                primes f    False     return  i for i in primes if primes i   True   print primes sieve1 2000000

User · Answer

Using a bit of numpy  I could find all primes below 100 million in a little over 2 seconds    There are two key features one should note   Cut out multiples of i only for i up to root of n Setting multiples of i to False using x 2 i  i    False is much faster than an explicit python for loop    These two significantly speed up your code  For limits below one million  there is no perceptible running time   import numpy as np  def primes n       x   np ones  n 1    dtype np bool      x 0    False     x 1    False     for i in range 2  int n  0 5  1           if x i               x 2 i  i    False      primes   np where x    True  0      return primes  print len primes 100 000 000

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I figured it must be possible to simply use the empty list as the terminating condition for the loop and came up with this   limit   100 ints   list range 2  limit       Will end up empty  while len ints   gt  0      prime   ints 0      print prime     ints remove prime      i   2     multiple   prime   i     while multiple  lt   limit          if multiple in ints              ints remove multiple          i    1         multiple   prime   i

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By combining contributions from many enthusiasts  including Glenn Maynard and MrHIDEn from above comments   I came up with following piece of code in python 2   def simpleSieve sieveSize        creating Sieve      sieve    True     sieveSize 1        0 and 1 are not considered prime      sieve 0    False     sieve 1    False     for i in xrange 2 int math sqrt sieveSize   1           if sieve i     False              continue         for pointer in xrange i  2  sieveSize 1  i               sieve pointer    False       Sieve is left with prime numbers    True     primes          for i in xrange sieveSize 1           if sieve i     True              primes append i      return primes  sieveSize   input   primes   simpleSieve sieveSize    Time taken for computation on my machine for different inputs in power of 10 is    3   0 3 ms 4   2 4 ms 5   23 ms 6   0 26 s 7   3 1 s 8   33 s

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import math def sieve n       primes    True  n     primes 0    False     primes 1    False     for i in range 2 int math sqrt n   1               j   i i             while j  lt  n                      primes j    False                     j   j i     return  x for x in range n  if primes x     True

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Much faster   import time def get primes n     m   n 1    numbers    True for i in range m     numbers    True    m  EDIT  faster   for i in range 2  int n  0 5   1        if numbers i         for j in range i i  m  i           numbers j    False   primes        for i in range 2  m       if numbers i         primes append i    return primes  start   time time   primes   get primes 10000  print time time   - start  print get primes 100

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A simple speed hack  when you define the variable  primes   set the step to 2 to skip all even numbers automatically  and set the starting point to 1   Then you can further optimize by instead of for i in primes  use for i in primes  round len primes     0 5    That will dramatically increase performance  In addition  you can eliminate numbers ending with 5 to further increase speed

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i think this is shortest code for finding primes with eratosthenes method  def prime r       n   range 2 r      while len n  gt 0          yield n 0          n    x for x in n if x not in range n 0  r n 0      print list prime r

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Probably the quickest way to have primary numbers is the following  import sympy list sympy primerange lower  upper 1    In case you don t need to store them  just use the code above without conversion to the list  sympy primerange is a generator  so it does not consume memory

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Removing from the beginning of an array  list  requires moving all of the items after it down   That means that removing every element from a list in this way starting from the front is an O n 2  operation   You can do this much more efficiently with sets   def primes sieve limit       limitn   limit 1     not prime   set       primes           for i in range 2  limitn           if i in not prime              continue          for f in range i 2  limitn  i               not prime add f           primes append i       return primes  print primes sieve 1000000        or alternatively  avoid having to rearrange the list   def primes sieve limit       limitn   limit 1     not prime    False    limitn     primes           for i in range 2  limitn           if not prime i               continue         for f in xrange i 2  limitn  i               not prime f    True          primes append i       return primes

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You re not quite implementing the correct algorithm   In your first example  primes sieve doesn t maintain a list of primality flags to strike unset  as in the algorithm   but instead resizes a list of integers continuously  which is very expensive  removing an item from a list requires shifting all subsequent items down by one   In the second example  primes sieve1 maintains a dictionary of primality flags  which is a step in the right direction  but it iterates over the dictionary in undefined order  and redundantly strikes out factors of factors  instead of only factors of primes  as in the algorithm   You could fix this by sorting the keys  and skipping non-primes  which already makes it an order of magnitude faster   but it s still much more efficient to just use a list directly   The correct algorithm  with a list instead of a dictionary  looks something like   def primes sieve2 limit       a    True    limit                            Initialize the primality list     a 0    a 1    False      for  i  isprime  in enumerate a           if isprime              yield i             for n in range i i  limit  i         Mark factors non-prime                 a n    False    Note that this also includes the algorithmic optimization of starting the non-prime marking at the prime s square  i i  instead of its double

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I just came up with this  It may not be the fastest  but I m not using anything other than straight additions and comparisons  Of course  what stops you here is the recursion limit  def nondivsby2        j   1     while True          j    2         yield j  def nondivsbyk k  nondivs       j   0     for i in nondivs          while j  lt  i              j    k         if j  gt  i              yield i  def primes        nd   nondivsby2       while True          p   next nd          nd   nondivsbyk p  nd          yield p  def main        for p in primes            print p

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Here s a version that s a bit more memory-efficient  and  a proper sieve  not trial divisions    Basically  instead of keeping an array of all the numbers  and crossing out those that aren t prime  this keeps an array of counters - one for each prime it s discovered - and leap-frogging them ahead of the putative prime   That way  it uses storage proportional to the number of primes  not up to to the highest prime   import itertools  def primes         class counter          def   init    this   n   this n  this current   this isVirgin   n  n n   True               isVirgin means it s never been incremented         def advancePast  this   n     return true if the counter advanced             if this current  gt  n                  if this isVirgin  raise StopIteration   if this is virgin  then so will be all the subsequent counters   Don t need to iterate further                  return False             this current    this n   pre  this current    n  post  this current  gt  n              this isVirgin   False   when it s gone  it s gone             return True      yield 1     multiples          for n in itertools count 2           isPrime   True         for p in  m advancePast n  for m in multiples               if p  isPrime   False         if isPrime              yield n             multiples append  counter  n     You ll note that primes   is a generator  so you can keep the results in a list or you can use them directly   Here s the first n primes   import itertools  for k in itertools islice  primes     n       print  k    And  for completeness  here s a timer to measure the performance   import time  def timer         t   k   time process time     10     for p in primes            if p gt k              print  time process time  -t     to     p     n               k    10             if k gt 100000  return   Just in case you re wondering  I also wrote primes   as a simple iterator  using   iter   and   next     and it ran at almost the same speed   Surprised me too

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not sure if my code is efficeient  anyone care to comment  from math import isqrt  def isPrime n       if n  gt   2    cheating the 2  is 2 even prime          for i in range 3  int n   2   1  2     dont waste time with even numbers             if n   i    0                  return False     return True  def primesTo n        x    2  if n  gt   2 else      cheat the only even prime     if n  gt   2          for i in range 3  n   1 2     dont waste time with even numbers             if isPrime i                   x append i        return x  def primes2 n     trying to do this using set methods and the  quot Sieve of Eratosthenes quot      base    2    again cheating the 2     base update set range 3  n   1  2      build the base of odd numbers     for i in range 3  isqrt n    1  2     apply the sieve         base difference update set range 2   i  n   1   i        return list base   print primesTo 10000     2 different methods for comparison print primes2 10000

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My implementation   import math n   100 marked      for i in range 2  int math sqrt n         if not marked get i           for x in range i   i  n  i               marked x    True  for i in range 2  n       if not marked get i           print i

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I prefer NumPy because of speed   import numpy as np    Find all prime numbers using Sieve of Eratosthenes def get primes1 n       m   int np sqrt n       is prime   np ones n  dtype bool      is prime  2    False    0 and 1 are not primes      for i in range 2  m           if is prime i     False              continue         is prime i i  i    False      return np nonzero is prime  0     Find all prime numbers using brute-force  def isprime n           Check if integer n is a prime         n   abs int n      n is a positive integer     if n  lt  2     0 and 1 are not primes         return False     if n    2     2 is the only even prime number         return True     if not n  amp  1     all other even numbers are not primes         return False       Range starts with 3 and only needs to go up the square root       of n for all odd numbers     for x in range 3  int n  0 5  1  2           if n   x    0              return False     return True    To apply a function to a numpy array  one have to vectorize the function def get primes2 n       vectorized isprime   np vectorize isprime      a   np arange n      return a vectorized isprime a     Check the output   n   100 print get primes1 n   print get primes2 n             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        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    Compare the speed of Sieve of Eratosthenes and brute-force on Jupyter Notebook  Sieve of Eratosthenes in 539 times faster than brute-force for million elements     timeit get primes1 1000000   timeit get primes2 1000000  4 79 ms    90 3   s per loop  mean    std  dev  of 7 runs  100 loops each  2 58 s    31 2 ms per loop  mean    std  dev  of 7 runs  1 loop each

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The fastest implementation I could come up with   isprime    True  N isprime 0    isprime 1    False for i in range 4  N  2       isprime i    False for i in range 3  N  2       if isprime i           for j in range i i  N  2 i               isprime j    False

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def eratosthenes n       multiples          for i in range 2  n 1           if i not in multiples              print  i              for j in range i i  n 1  i                   multiples append j   eratosthenes 100

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I realise this isn t really answering the question of how to generate primes quickly  but perhaps some will find this alternative interesting  because python provides lazy evaluation via generators  eratosthenes  sieve can be implemented exactly as stated   def intsfrom n       while True          yield n         n    1  def sieve ilist       p   next ilist      yield p     for q in sieve n for n in ilist if n p    0           yield q   try      for p in sieve intsfrom 2            print p       print    except RuntimeError as e      print e   The try block is there because the algorithm runs until it blows the stack and without the try block the backtrace is displayed pushing the actual output you want to see off screen

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Using recursion and walrus operator  def prime factors n       for i in range 2  int n    0 5    1           if  q r    divmod n  i   1     0              return  i    factor list q r 0       return  n

[python] Sieve of Eratosthenes - Finding Primes Python

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