Posts Tagged ‘eratosthenes’

Programming Praxis – Sieve Of Euler

February 25, 2011

In today’s Programming Praxis exercise, our goal is to implement the Sieve of Euler to generate primes. Let’s get started, shall we?

Some imports:

import Data.List.Ordered
import Data.Time.Clock

Euler’s sieve is pretty simple: just filter out all remaining multiples of each successive prime number. We add the minor optimization of only considering odd numbers.

primes_euler :: [Integer]
primes_euler = 2 : euler [3,5..] where
    euler ~(x:xs) = x : euler (minus xs $ map (* x) (x:xs))

To compare, we use the Sieve of Eratosthenes implementation found here:

primes_eratosthenes :: [Integer]
primes_eratosthenes = 2 : 3 : sieve (tail primes_eratosthenes) 3 [] where
    sieve ~(p:ps) x fs = let q=p*p in
        foldr (flip minus) [x+2,x+4..q-2] [[y+s,y+2*s..q] | (s,y) <- fs]
        ++ sieve ps q ((2*p,q):[(s,q-rem (q-y) s) | (s,y) <- fs])

A quick and dirty benchmarking function:

benchPrimes :: Num a => [a] -> IO ()
benchPrimes f = do start <- getCurrentTime
                   print . sum $ take 10000 f
                   print . flip diffUTCTime start =<< getCurrentTime

Time to run the benchmarks.

main :: IO ()
main = do benchPrimes primes_eratosthenes
          benchPrimes primes_euler

The sieve of Eratosthenes runs in about 0.02 seconds, while Euler’s sieve takes around 6.6 seconds, which means this implementation of Euler’s Sieve is about 300-400 times slower. Clearly you don’t want to use this version in practice.