Is there a generic way to memoize in Scala?

Question

I wanted to memoize this:

def fib(n: Int) = if(n <= 1) 1 else fib(n-1) + fib(n-2)
println(fib(100)) // times out

So I wrote this and this surprisingly compiles and works (I am surprised because fib references itself in its declaration):

case class Memo[A,B](f: A => B) extends (A => B) {
  private val cache = mutable.Map.empty[A, B]
  def apply(x: A) = cache getOrElseUpdate (x, f(x))
}

val fib: Memo[Int, BigInt] = Memo {
  case 0 => 0
  case 1 => 1
  case n => fib(n-1) + fib(n-2) 
}

println(fib(100))     // prints 100th fibonacci number instantly

But when I try to declare fib inside of a def, I get a compiler error:

def foo(n: Int) = {
  val fib: Memo[Int, BigInt] = Memo {
    case 0 => 0
    case 1 => 1
    case n => fib(n-1) + fib(n-2) 
  }
  fib(n)
} 

Above fails to compile error: forward reference extends over definition of value fib case n => fib(n-1) + fib(n-2)

Why does declaring the val fib inside a def fails but outside in the class/object scope works?

To clarify, why I might want to declare the recursive memoized function in the def scope - here is my solution to the subset sum problem:

/**
   * Subset sum algorithm - can we achieve sum t using elements from s?
   *
   * @param s set of integers
   * @param t target
   * @return true iff there exists a subset of s that sums to t
   */
  def subsetSum(s: Seq[Int], t: Int): Boolean = {
    val max = s.scanLeft(0)((sum, i) => (sum + i) max sum)  //max(i) =  largest sum achievable from first i elements
    val min = s.scanLeft(0)((sum, i) => (sum + i) min sum)  //min(i) = smallest sum achievable from first i elements

    val dp: Memo[(Int, Int), Boolean] = Memo {         // dp(i,x) = can we achieve x using the first i elements?
      case (_, 0) => true        // 0 can always be achieved using empty set
      case (0, _) => false       // if empty set, non-zero cannot be achieved
      case (i, x) if min(i) <= x && x <= max(i) => dp(i-1, x - s(i-1)) || dp(i-1, x)  // try with/without s(i-1)
      case _ => false            // outside range otherwise
    }

    dp(s.length, t)
  }

Answer 1

I found a better way to memoize using Scala:

def memoize[I, O](f: I => O): I => O = new mutable.HashMap[I, O]() {
  override def apply(key: I) = getOrElseUpdate(key, f(key))
}

Now you can write fibonacci as follows:

lazy val fib: Int => BigInt = memoize {
  case 0 => 0
  case 1 => 1
  case n => fib(n-1) + fib(n-2)
}

Here's one with multiple arguments (the choose function):

lazy val c: ((Int, Int)) => BigInt = memoize {
  case (_, 0) => 1
  case (n, r) if r > n/2 => c(n, n - r)
  case (n, r) => c(n - 1, r - 1) + c(n - 1, r)
}

And here's the subset sum problem:

// is there a subset of s which has sum = t
def isSubsetSumAchievable(s: Vector[Int], t: Int) = {
  // f is (i, j) => Boolean i.e. can the first i elements of s add up to j
  lazy val f: ((Int, Int)) => Boolean = memoize {
    case (_, 0) => true        // 0 can always be achieved using empty list
    case (0, _) => false       // we can never achieve non-zero if we have empty list
    case (i, j) => 
      val k = i - 1            // try the kth element
      f(k, j - s(k)) || f(k, j)
  }
  f(s.length, t)
}

EDIT: As discussed below, here is a thread-safe version

def memoize[I, O](f: I => O): I => O = new mutable.HashMap[I, O]() {self =>
  override def apply(key: I) = self.synchronized(getOrElseUpdate(key, f(key)))
}

Answer 2

Class/trait level val compiles to a combination of a method and a private variable. Hence a recursive definition is allowed.

Local vals on the other hand are just regular variables, and thus recursive definition is not allowed.

By the way, even if the def you defined worked, it wouldn't do what you expect. On every invocation of foo a new function object fib will be created and it will have its own backing map. What you should be doing instead is this (if you really want a def to be your public interface):

private val fib: Memo[Int, BigInt] = Memo {
  case 0 => 0
  case 1 => 1
  case n => fib(n-1) + fib(n-2) 
}

def foo(n: Int) = {
  fib(n)
} 

Answer 3

Scalaz has a solution for that, why not reuse it?

import scalaz.Memo
lazy val fib: Int => BigInt = Memo.mutableHashMapMemo {
  case 0 => 0
  case 1 => 1
  case n => fib(n-2) + fib(n-1)
}

You can read more about memoization in Scalaz.

Answer 4

Mutable HashMap isn't thread safe. Also defining case statements separately for base conditions seems unnecessary special handling, rather Map can be loaded with initial values and passed to Memoizer. Following would be the signature of Memoizer where it accepts a memo(immutable Map) and formula and returns a recursive function.

Memoizer would look like

def memoize[I,O](memo: Map[I, O], formula: (I => O, I) => O): I => O

Now given a following Fibonacci formula,

def fib(f: Int => Int, n: Int) = f(n-1) + f(n-2)

fibonacci with Memoizer can be defined as

val fibonacci = memoize( Map(0 -> 0, 1 -> 1), fib)

where context agnostic general purpose Memoizer is defined as

    def memoize[I, O](map: Map[I, O], formula: (I => O, I) => O): I => O = {
        var memo = map
        def recur(n: I): O = {
          if( memo contains n) {
            memo(n) 
          } else {
            val result = formula(recur, n)
            memo += (n -> result)
            result
          }
        }
        recur
      }

Similarly, for factorial, a formula is

def fac(f: Int => Int, n: Int): Int = n * f(n-1)

and factorial with Memoizer is

val factorial = memoize( Map(0 -> 1, 1 -> 1), fac)

Inspiration: Memoization, Chapter 4 of Javascript good parts by Douglas Crockford