Finding primes using a primes map with Haskell and Hugs98
Wed, 20 Dec 2000 16:02:23 +0200 (IST)
On Tue, 19 Dec 2000, Simon Peyton-Jones wrote:
> | Another way to do this is to compute the final array directly,
> | instead of computing successive versions of the array:
> | import Array
> | primes n = [ i | i <- [2 ..n], not (primesMap ! i)] where
> | primesMap = accumArray (||) False (2,n) multList
> | multList = [(m,True) | j <- [2 .. n `div` 2], m <-
> | multiples j]
> | multiples j = takeWhile (n>=) [k*j | k <- [2..]]
> This style is definitely the way to go. Haskell does badly
> if you update an array one index at a time.
Unfortunately, it seems that this style is not the way to go. This program
cannot scale beyond 5000 while my second program scales beyond 30000. I'm
not saying 30000 is a good limit, but 5000 is much worse.
Anyway, somebody who contacted me in private suggested the following
method. It is a similiar algorithm which uses a list instead of an array.
primes :: Int -> [Int]
primes how_much = sieve [2..how_much] where
sieve (p:x) =
p : (if p <= mybound
then sieve (remove (p*p) x)
else x) where
remove what (a:as) | what > how_much = (a:as)
| a < what = a:(remove what as)
| a == what = (remove (what+step) as)
| a > what = a:(remove (what+step) as)
remove what  = 
step = (if (p == 2) then p else (2*p))
sieve  = 
mybound = ceiling(sqrt(fromIntegral how_much))
I optimized it quite a bit, but the concept remained the same.
Anyway, this code can scale very well to 100000 and beyond. But it's not
exactly the same algorithm.
I also implemented this algorithm in perl, and I can send it in person if
anybody requests it.
I'll try to see how the two programs run in GHC and HBC.
> Remember that arrays can be recursive. Here's a definition
> of Fibonacci for example; you can probably adapt it for primes
> fibs :: Int -> Array Int Int
> -- If a = fibs n, then a!i is fib(i), for i<=n.
> fibs n = a
> a = array (1,n) ([(1,1),(2,1)] ++ [(i,a!(i-1) + a!(i-2) | i <-
> -- Notice that a is recursive
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