performance - Why is this scala prime generation so slow/memory intensive?

Question

I run out of memory while finding the 10,001th prime number.

object Euler0007 {
  def from(n: Int): Stream[Int] = n #:: from(n + 1)
  def sieve(s: Stream[Int]): Stream[Int] = s.head #:: sieve(s.filter(_ % s.head != 0))
  def primes = sieve(from(2))
  def main(args: Array[String]): Unit = {
    println(primes(10001))
  }
}

Is this because after each "iteration" (is this the correct term in this context?) of primes, I increase the stack of functions to be called to get the next element by one?

One solution that I've found on the web which doesn't resort to an iterative solution (which I'd like to avoid to get into functional programming/idiomatic scala) is this (Problem 7):

lazy val ps: Stream[Int] = 2 #:: Stream.from(3).filter(i => ps.takeWhile(j => j * j <= i).forall(i % _ > 0))

From what I can see, this does not lead to this recursion-like way. Is this a good way to do it, or do you know of a better way?

Answer 1

One reason why this is slow is that it isn't the sieve of Eratosthenes. Read http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf for a detailled explanation (the examples are in Haskell, but can be translated directly into Scala).

My old solution for Euler problem #7 wasn't the "true" sieve either, but it seems to work good enough for little numbers:

object Sieve {

    val primes = 2 #:: sieve(3)

    def sieve(n: Int) : Stream[Int] =
          if (primes.takeWhile(p => p*p <= n).exists(n % _ == 0)) sieve(n + 2)
          else n #:: sieve(n + 2)

    def main(args: Array[String]) {
      println(primes(10000)) //note that indexes are zero-based
    }
}

I think the problem with your first version is that you have only defs and no val which collects the results and can be consulted by the generating function, so you always recalculate from scratch.