## Posts Tagged ‘sieve’

### Programming Praxis – Sieving For Totients

July 10, 2012

In today’s Programming Praxis exercise, our goal is to calculate the totients of a given range of numbers using a sieve. Let’s get started, shall we?

Due to the way I structured my code, Data.Map is a little more convenient than Data.Vector (since Data.Vector lacks the equivalent of the adjust function).

`import qualified Data.Map as M`

The sieving can be solved easily with two folds. The outer one to check all the elements in the list, and the inner one to update all the multiples of a given index. One space-saving trick is to realize that you don’t need to treat i and its multiples differently, since i * (1 – 1/i) = i – i/i = i – 1. This saves a separate insert call.

```totients :: Integral a => a -> [a]
totients n = M.elems \$ foldl (\m i -> if m M.! i == i
then foldr (M.adjust (\x -> div (x*(i-1)) i)) m [i,2*i..n] else m)
(M.fromList \$ zip [0..n] [0..n]) [2..n]```

A test to see if everything is working properly:

```main :: IO ()
main = print \$ totients 100```

### 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.