Sieve of Eratosthenes in Haskell

Question

I'm solving some classic problems in Haskell to develop my functional skills and I have a problem to implement an optimization suggested at this "Programming Praxis" site:

I have three solutions to this problem and the third one is too slow compared to the second solution. Can someone suggest some improvements to my code?

My implementations are:

-- primeira implementação
primes n
    | n < 2 = []
    | n == 2 = [2]
    | n `mod` 2 == 0 = primes'
    | otherwise = if (find (\x -> n `mod` x == 0) primes') == Nothing then
                      n:primes'
                  else
                      primes'
    where primes' = primes (n - 1)

-- segunda implementação
primes' :: Integer -> [Integer]
primes' n = sieve $ 2 : [3,5..n]
    where sieve :: [Integer] -> [Integer]
          sieve [] = []
          sieve l@(x:xs)
              | x*x >= n = l
              | otherwise = x : sieve list'
              where list' = filter (\y -> y `mod` x /= 0) xs

-- terceira implementação
primes'' :: Integer -> [Integer]
primes'' n = 2 : sieve 3 [3,5..n]
    where sieve :: Integer -> [Integer] -> [Integer]
          sieve _ [] = []
          sieve m l@(x:xs)
              | m*m >= n = l
              | x < m*m = x : sieve m xs
              | otherwise = sieve (m + 2) list'
              where list'= filter (\y -> y `mod` m /= 0) l

Answer 1

Looks to me like the problem with your third revision is how you choose the next element to sift on. You indiscriminately increment by 2. The problem is that you then sift on unnecessary numbers. for example, in this version your eventually going to pass 9 as m, and you're going to do an extra recursion to filter on 9, even though it isn't even in the list, and thus you should have never picked it in the first place (since it would have been removed in the very first filter on 3)

Even though the second version doesn't start the filtering past the square of the number it sifts on, it never chooses an unnecessary sifting value.

In other words, I think you end up sifting on every odd number between 3 and n. Instead you should be sifting on every odd number that hasn't already been removed by a previous pass.

I think to correctly implement the optimization of starting the sieve at the square of the current sift value, you have to retain the front of the list while sifting on the back where back contains the elements >= the square of the sift value. I think this would force you to use concatenations, and I'm not so sure that the optimization is good enough to cancel out the overhead induced by using ++.

Answer 2

First of all, mod is slow so use rem in situations where it doesn't matter (when you aren't dealing with negatives, basically). Secondly, use Criterion to show (to yourself) what is faster and what changes are actually optimizations. I know I'm not giving a full answer to you question with this, but its a good place for you (and other potential answerers) to start, so here's some code:

import List
import Criterion.Main

main = do
  str <- getLine
  let run f = length . f
      input = read str :: Integer
  defaultMain   [ bench "primes" (nf (run primes) input)
                , bench "primes'" (nf (run primes') input)
                , bench "primes''" (nf (run primes'') input)
                , bench "primesTMD" (nf (run primesTMD) input)
                , bench "primes'TMD" (nf (run primes'TMD) input)
                , bench "primes''TMD" (nf (run primes''TMD) input)
                ]
  putStrLn . show . length . primes'' $ (read str :: Integer)

-- primeira implementação
primes n
    | n < 2 = []
    | n == 2 = [2]
    | n `mod` 2 == 0 = primes'
    | otherwise = if (find (\x -> n `mod` x == 0) primes') == Nothing then
                      n:primes'
                  else
                      primes'
    where primes' = primes (n - 1)

primesTMD n
    | n < 2 = []
    | n == 2 = [2]
    | n `mod` 2 == 0 = primes'
    | otherwise = if (find (\x -> n `rem` x == 0) primes') == Nothing then
                      n:primes'
                  else
                      primes'
    where primes' = primesTMD (n - 1)

-- segunda implementação
primes' :: Integer -> [Integer]
primes' n = sieve $ 2 : [3,5..n]
    where sieve :: [Integer] -> [Integer]
          sieve [] = []
          sieve l@(x:xs)
              | x*x >= n = l
              | otherwise = x : sieve list'
              where list' = filter (\y -> y `mod` x /= 0) xs

primes'TMD :: Integer -> [Integer]
primes'TMD n = sieve $ 2 : [3,5..n]
    where sieve :: [Integer] -> [Integer]
          sieve [] = []
          sieve l@(x:xs)
              | x*x >= n = l
              | otherwise = x : sieve list'
              where list' = filter (\y -> y `rem` x /= 0) xs

-- terceira implementação
primes'' :: Integer -> [Integer]
primes'' n = 2 : sieve 3 [3,5..n]
    where sieve :: Integer -> [Integer] -> [Integer]
          sieve _ [] = []
          sieve m l@(x:xs)
              | m*m >= n = l
              | x < m*m = x : sieve m xs
              | otherwise = sieve (m + 2) list'
              where list'= filter (\y -> y `mod` m /= 0) l

primes''TMD :: Integer -> [Integer]
primes''TMD n = 2 : sieve 3 [3,5..n]
    where sieve :: Integer -> [Integer] -> [Integer]
          sieve _ [] = []
          sieve m l@(x:xs)
              | m*m >= n = l
              | x < m*m = x : sieve m xs
              | otherwise = sieve (m + 2) list'
              where list'= filter (\y -> y `rem` m /= 0) l

Notice the improved runtime of the variants using rem:

 $ ghc --make -O2 sieve.hs
 $./sieve
 5000
 ...
 benchmarking primes 
 mean: 23.88546 ms, lb 23.84035 ms, ub 23.95000 ms

 benchmarking primes'
 mean: 775.9981 us, lb 775.4639 us, ub 776.7081 us

 benchmarking primes''
 mean: 837.7901 us, lb 836.7824 us, ub 839.0260 us

 benchmarking primesTMD
 mean: 16.15421 ms, lb 16.11955 ms, ub 16.19202 ms

 benchmarking primes'TMD
 mean: 568.9857 us, lb 568.5819 us, ub 569.4641 us

 benchmarking primes''TMD
 mean: 642.5665 us, lb 642.0495 us, ub 643.4105 us

While I see you are doing this for your own education, its worth noting the related links of Primes on Haskell.org and the fast Primes package on hackage.

Answer 3

This is not optimized but expressive implementation: check video Sieve of Eratosthenes in haskell

import qualified Data.Set as Set(fromList,difference)
kr n l = (*n) <$> [2..l `div` n]
g n = difference (fromList [2..n]) (fromList $ concat $ ((flip kr) n) <$> [2..n])