Reduce, fold or scan (Left/Right)?

Question

When should I use reduceLeft, reduceRight, foldLeft, foldRight, scanLeft or scanRight?

I want an intuition/overview of their differences - possibly with some simple examples.

Thanks for the pointer. It's a bit above my technical knowledge :) Is there something in my answer that you think should be clarified/changed? — Marc Grue
No, just pointing out a bit of history and the relevance to MPP. — samthebest
Well, strictly speaking the distinction between reduce and fold is NOT the existence of a start value - rather that is a consequence of a more deep underlying mathematical reason. — samthebest

Marc Grue Marc Grue · Accepted Answer · 2013-07-01T16:04:30

In general, all 6 fold functions apply a binary operator to each element of a collection. The result of each step is passed on to the next step (as input to one of the binary operator's two arguments). This way we can cumulate a result.

reduceLeft and reduceRight cumulate a single result.

foldLeft and foldRight cumulate a single result using a start value.

scanLeft and scanRight cumulate a collection of intermediate cumulative results using a start value.

Accumulate

From LEFT and forwards...

With a collection of elements abc and a binary operator add we can explore what the different fold functions do when going forwards from the LEFT element of the collection (from A to C):

val abc = List("A", "B", "C")

def add(res: String, x: String) = { 
  println(s"op: $res + $x = ${res + x}")
  res + x
}

abc.reduceLeft(add)
// op: A + B = AB
// op: AB + C = ABC    // accumulates value AB in *first* operator arg `res`
// res: String = ABC

abc.foldLeft("z")(add) // with start value "z"
// op: z + A = zA      // initial extra operation
// op: zA + B = zAB
// op: zAB + C = zABC
// res: String = zABC

abc.scanLeft("z")(add)
// op: z + A = zA      // same operations as foldLeft above...
// op: zA + B = zAB
// op: zAB + C = zABC
// res: List[String] = List(z, zA, zAB, zABC) // maps intermediate results

From RIGHT and backwards...

If we start with the RIGHT element and go backwards (from C to A) we'll notice that now the second argument to our binary operator accumulates the result (the operator is the same, we just switched the argument names to make their roles clear):

def add(x: String, res: String) = {
  println(s"op: $x + $res = ${x + res}")
  x + res
}

abc.reduceRight(add)
// op: B + C = BC
// op: A + BC = ABC  // accumulates value BC in *second* operator arg `res`
// res: String = ABC

abc.foldRight("z")(add)
// op: C + z = Cz
// op: B + Cz = BCz
// op: A + BCz = ABCz
// res: String = ABCz

abc.scanRight("z")(add)
// op: C + z = Cz
// op: B + Cz = BCz
// op: A + BCz = ABCz
// res: List[String] = List(ABCz, BCz, Cz, z)

.

De-cumulate

From LEFT and forwards...

If instead we were to de-cumulate some result by subtraction starting from the LEFT element of a collection, we would cumulate the result through the first argument res of our binary operator minus:

val xs = List(1, 2, 3, 4)

def minus(res: Int, x: Int) = {
  println(s"op: $res - $x = ${res - x}")
  res - x
}

xs.reduceLeft(minus)
// op: 1 - 2 = -1
// op: -1 - 3 = -4  // de-cumulates value -1 in *first* operator arg `res`
// op: -4 - 4 = -8
// res: Int = -8

xs.foldLeft(0)(minus)
// op: 0 - 1 = -1
// op: -1 - 2 = -3
// op: -3 - 3 = -6
// op: -6 - 4 = -10
// res: Int = -10

xs.scanLeft(0)(minus)
// op: 0 - 1 = -1
// op: -1 - 2 = -3
// op: -3 - 3 = -6
// op: -6 - 4 = -10
// res: List[Int] = List(0, -1, -3, -6, -10)

From RIGHT and backwards...

But look out for the xRight variations now! Remember that the (de-)cumulated value in the xRight variations is passed to the second parameter res of our binary operator minus:

def minus(x: Int, res: Int) = {
  println(s"op: $x - $res = ${x - res}")
  x - res
}

xs.reduceRight(minus)
// op: 3 - 4 = -1
// op: 2 - -1 = 3  // de-cumulates value -1 in *second* operator arg `res`
// op: 1 - 3 = -2
// res: Int = -2

xs.foldRight(0)(minus)
// op: 4 - 0 = 4
// op: 3 - 4 = -1
// op: 2 - -1 = 3
// op: 1 - 3 = -2
// res: Int = -2

xs.scanRight(0)(minus)
// op: 4 - 0 = 4
// op: 3 - 4 = -1
// op: 2 - -1 = 3
// op: 1 - 3 = -2
// res: List[Int] = List(-2, 3, -1, 4, 0)

The last List(-2, 3, -1, 4, 0) is maybe not what you would intuitively expect!

As you see, you can check what your foldX is doing by simply running a scanX instead and debug the cumulated result at each step.

Bottom line

Cumulate a result with reduceLeft or reduceRight.
Cumulate a result with foldLeft or foldRight if you have a start value.
Cumulate a collection of intermediate results with scanLeft or scanRight.
Use a xLeft variation if you want to go forwards through the collection.
Use a xRight variation if you want to go backwards through the collection.

Reduce, fold or scan (Left/Right)?

3 Answers

Accumulate

De-cumulate

Bottom line

In conclusion