Lorenzo Bolla lbolla at gmail.com
Wed Sep 12 11:26:46 CEST 2012

```On Wed, Sep 12, 2012 at 9:06 AM, Yi Cheng <chengyidna at gmail.com> wrote:

> Recently, I'm trying to solve some problems in project euler using
> haskell. When it came to problem 10, calculating the sum of all primes
> below 20000000, I try to write a program which can generate primes.
> In my memory Eratosthenes is faster than just whether a number can be
> divided by the number less then the square root of it.
> Firstly, I wrote the following programs:
>
> module Main where
> isPrime x = isPrime' 3 x (round . sqrt. fromIntegral \$ x)
> isPrime' d target maxd
>   | d > maxd = True
>   | mod target d == 0 = False
>   | otherwise = isPrime' (d + 2) target maxd
>
> main = print \$ (sum (filter isPrime [3,5..2000000]) + 2)
>
> And it consume about 11s in my computer.
> Then, I tried to figure out how to solve the problem by Eratosthenes, but
> failed. Later, I find a program implemented by others, meeting my purpose
> and I've used it to solve the problem:
>
> primes :: [Int]
> primes = primes' [2..]
>
> primes' :: [Int] -> [Int]
> primes' [] = []
> primes' (n:ns) = n : primes' (filter (\v -> v `mod` n /= 0) ns)
>
> solve x = sum \$ primes' [2..x]
>
> main = print \$ solve 2000000
>
> Well, although the code is beautiful, it is slow. Even waiting for a
> minute, no answer was printed.
>
> In C version, Eratosthenes is faster than the method implemented in my
> earlier code, which only consume 0.3s(the earlier method consume 1.6s).
>
> So I want to know, why Eratosthenes implemented in Haskell is slow than
> the ugly code implemented by me.
> Could anyone tell me?
>
>
Eratosthenes's complexity is O(n^2) (both space and time), whereas the
"ugly" one has a sub-quadratic running complexity and linear in space.

Try to profile them:
\$> ghc -O2 --make -prof -auto-all <filename>
\$> ./primes +RTS -p -hc
\$> hp2ps primes.hp

You'll see that most of the time is spent allocating space which is never
released.
You could play a bit with strictness, but the main problem is the awful
complexity of the algorithm.

hth,
L.

> Thank you
> Yi Cheng
>
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