Lazy Evaluation

Motivation

Consider the following program that finds the second prime number between 1000 and 10000:

((1000 to 10000) filter isPrime)(1)

This is much shorter than the recursive alternative:

def secondPrime(from: Int, to: Int) = nthPrime(from, to, 2)
def nthPrime(from: Int, to: Int, n: Int): Int =
  if (from >= to) throw new Error("no prime")
  else if (isPrime(from))
    if (n == 1) from else nthPrime(from + 1, to, n - 1)
  else nthPrime(from + 1, to, n)

But from a standpoint of performance, the first version is pretty bad; it constructs all prime numbers between 1000 and 10000 in a list, but only ever looks at the first two elements of that list.

Reducing the upper bound would speed things up, but risks that we miss the second prime number all together.

Delayed Evaluation

However, we can make the short-code efficient by using a trick:

  • Avoid computing the tail of a sequence until it is needed for the evaluation result (which might be never)

This idea is implemented in a new class, the LazyList.

LazyLists are similar to lists, but their elements are evaluated only on demand.

Defining LazyLists

LazyLists are defined from a constructor LazyList.cons.

For instance,

val xs = LazyList.cons(1, LazyList.cons(2, LazyList.empty))

LazyList Ranges

Let's try to write a function that returns a LazyList representing a range of numbers between lo and hi:

def llRange(lo: Int, hi: Int): LazyList[Int] =
  if (lo >= hi) LazyList.empty
  else LazyList.cons(lo, llRange(lo + 1, hi))

Compare to the same function that produces a list:

def listRange(lo: Int, hi: Int): List[Int] =
  if (lo >= hi) Nil
  else lo :: listRange(lo + 1, hi)

The functions have almost identical structure yet they evaluate quite differently.

  • listRange(start, end) will produce a list with end - start elements and return it.
  • llRange(start, end) returns a single object of type LazyList with start as head element.
    • The other elements are only computed when they are needed, where “needed” means that someone calls tail on the stream.

Methods on LazyLists

LazyList supports almost all methods of List.

For instance, to find the second prime number between 1000 and 10000:

(llRange(1000, 10000) filter isPrime)(1)

The one major exception is ::.

x :: xs always produces a list, never a lazy list.

There is however an alternative operator #:: which produces a lazy list.

x #:: xs == LazyList.cons(x, xs)

#:: can be used in expressions as well as patterns.

Implementation of LazyLists

The implementation of lazy lists is quite close to the one of lists.

Here's the class LazyList:

final class LazyList[+A] ... extends ... {
  override def isEmpty: Boolean = ...
  override def head: A = ...
  override def tail: LazyList[A] = ...
  ...
}

As for lists, all other methods can be defined in terms of these three.

Concrete implementations of streams are defined in the LazyList.State companion object. Here's a first draft:

private object State {
  object Empty extends State[Nothing] {
    def head: Nothing = throw new NoSuchElementException("head of empty lazy list")
    def tail: LazyList[Nothing] = throw new UnsupportedOperationException("tail of empty lazy list")
  }

  final class Cons[A](val head: A, val tail: LazyList[A]) extends State[A]
}

The only important difference between the implementations of List and LazyList concern tail, the second parameter of LazyList.cons.

For lazy lists, this is a by-name parameter: the type of tail starts with =>. In such a case, this parameter is evaluated by following the rules of the call-by-name model.

That's why the second argument to LazyList.cons is not evaluated at the point of call.

Instead, it will be evaluated each time someone calls tail on a LazyList object.

In Scala 2.13, LazyList (previously Stream) became fully lazy from head to tail. To make it possible, methods (filter, flatMap...) are implemented in a way where the head is not being evaluated if is not explicitly indicated.

For instance, here's filter:

object LazyList extends SeqFactory[LazyList] {
  ...
  private def filterImpl[A](ll: LazyList[A], p: A => Boolean, isFlipped: Boolean): LazyList[A] = {
  // DO NOT REFERENCE `ll` ANYWHERE ELSE, OR IT WILL LEAK THE HEAD
  var restRef = ll                         // val restRef = new ObjectRef(ll)
  newLL {
    var elem: A = null.asInstanceOf[A]
    var found   = false
    var rest    = restRef                  // var rest = restRef.elem
    while (!found && !rest.isEmpty) {
      elem    = rest.head
      found   = p(elem) != isFlipped
      rest    = rest.tail
      restRef = rest                       // restRef.elem = rest
    }
    if (found) sCons(elem, filterImpl(rest, p, isFlipped)) else State.Empty
  }
}

Exercise

Consider the following modification of llRange. When you write llRange(1, 10).take(3).toList what is the value of rec?

Be careful, head is evaluating too!

var rec = 0
def llRange(lo: Int, hi: Int): LazyList[Int] = {
  rec = rec + 1
  if (lo >= hi) LazyList.empty
  else LazyList.cons(lo, llRange(lo + 1, hi))
}
llRange(1, 10).take(3).toList
rec shouldBe res0

Lazy Evaluation

The proposed LazyList implementation suffers from a serious potential performance problem: If tail is called several times, the corresponding stream will be recomputed each time.

This problem can be avoided by storing the result of the first evaluation of tail and re-using the stored result instead of recomputing tail.

This optimization is sound, since in a purely functional language an expression produces the same result each time it is evaluated.

We call this scheme lazy evaluation (as opposed to by-name evaluation in the case where everything is recomputed, and strict evaluation for normal parameters and val definitions.)

Lazy Evaluation in Scala

Haskell is a functional programming language that uses lazy evaluation by default.

Scala uses strict evaluation by default, but allows lazy evaluation of value definitions with the lazy val form:

lazy val x = expr

Exercise

val builder = new StringBuilder

val x = { builder += 'x'; 1 }
lazy val y = { builder += 'y'; 2 }
def z = { builder += 'z'; 3 }

z + y + x + z + y + x

builder.result() shouldBe res0