Reduction in Force

Our last talk about »== was full of twists and turns, some philosophy, surprising connections, and a radical new operator. It was a lot to absorb, and you may have to play with it some in your own code before you really know what it’s about. That’s ok.

Let’s take a little break and talk about a handy functional tool built into Swift stdlib. I promise no big reveals, no new operators, no fancy types; just hands-on, practical discussion of the Swiss Army knife of transform functions: reduce.

Reducing is almost easier to show than to explain. Let’s start with a trivial example. You have an array of numbers and you want to know the sum. We might say you want to “reduce” a [Int] to a single Int. reduce takes a sequence, an intial value, and a “combining” function like this:

let xs = [1,2,3,4,5]
let sum = reduce(xs, 0) { accumulator, value in accumulator + value }
// ==> 15

The most interesting part is the combining function. It is passed a “value so far” and a “next value from the list” and it needs to return the result of combining them. Its signature looks like this:

(U, T) -> U

So it doesn’t have to return the same kind of thing as is in the list, but it often does. In our example, since we want the sum, we just add the two values. We could just as easily reduce the list to its product:

let product = reduce(xs, 1) { accumulator, value in accumulator * value }
// ==> 120

I’m spelling out parameters to make it clear exactly what is being passed to the combining function, but we can of course use Swift shortcuts for this:

let sum = reduce(xs, 0) { $0 + $1 }

Or we can go futher, and shorten it to:

let sum = reduce(xs, 0, +)

What’s up with that? I told you before, + is just a function:

func +(lhs: Float, rhs: Float) -> Float

And it’s in the form (T, T) -> T, which is just a special case of (U, T) -> U. So that’s fine, and we can pass it as the combining function.

There’s also a method form that can be convenient:

let sum = xs.reduce(0, combine: +)

We can think of that as “take the list of x values, and add them, starting with zero.”

A note on origami and names

Another common name for this function is “fold.” Sometimes, in other languages, you’ll see “fold left” or “fold right.” Swift’s “reduce” is the same as “fold left.” (A right fold starts at the end of the list and goes backwards.1) You may also see it called “aggregate” or “accumulate” or “inject.” They’re all pretty much the same function.

Finding a starting point

OK, so we can add and multiply stuff. What else you got? We can get minimums and maximums. Let’s implement stdlib’s minElement for arrays with reduce.2

Reducing requires an initial value. So, what should it be? For types that have a known minimum (like Int.min), you might be tempted to use it. But that’s not very flexible, and some orderable types don’t have an obvious minimum. For example, what if we implemened a BigNum type that supported arbitrarily large or small values?

A common solution is to treat the first element as the initial value, and then reduce the rest:

func minElement<T: Comparable>(xs: [T]) -> T {
  return dropFirst(xs).reduce(xs[0], combine: min)

That’s a reusable pattern, so we could, if we wanted to, extract it:

extension Array {
  func reduce1(combine: (T, T) -> T) -> T {
    return dropFirst(self).reduce(self[0], combine: combine)

func minElement<T: Comparable>(xs: [T]) -> T {
  return xs.reduce1(min)

Notice that reduce1 requires a combining function that takes and returns the same type. It’s worth taking a moment and thinking about why that has to be true.

Of course, reduce1 crashes if the array is empty. What if we wanted to get back an Int? to guard against that? Here’s one way we could do it:

func safeMinElement<T: Comparable>(xs: [T]) -> T? {
  return xs.reduce(nil) { m, x in min(m ?? x, x) }

So we start with nil, and for each element if the current minimum is nil, we take whatever was passed in, and otherwise we take the minimum. Notice how we rely on Swift’s habit of promiting anything into an optional when required. Let’s us be pretty sloppy expressive.

Beyond math

The combining function (U, T) -> U says nothing about numbers. We can reduce anything we can imagine. We could join strings just as easily.

func join(elements: [String]) -> String {
  return elements.reduce("", +)

Or, as promised, here’s flatten, which takes a nested array and removes one level of structure:

func flatten<T>(xs: [[T]]) -> [T] {
  return xs.reduce([], +)

Told you it was simple. Notice how summing numbers, joining strings, and flattening arrays all have the same implementation, differing only in their initial value? That’s another thing worth thinking about for a while, but it’s too much theory for this post. (If you want to research it, the magic search term you want is monoid.)

Mapping an array is just a special case of reducing:3

func map<T, U>(xs: [T], f: T -> U) -> [U] {
  return xs.reduce([]) { $0 + [f($1)] }

So is filtering:

func filter<T>(xs: [T], includeElement: T -> Bool) -> [T] {
  return xs.reduce([]) { filtered, x in includeElement(x) ? filtered + [x] : filtered }


func flatMap<T, U>(xs: [T], f: T -> [U]) -> [U] {
  return xs.reduce([]) { $0 + f($1) }


func reverse<T>(xs: [T]) -> [T] {
  return reduce(xs, []) { [$1] + $0 }

Even counting!

func countElements<T>(xs: [T]) -> Int {
  return reduce(xs, 0) { c, _ in c + 1 }

You can do a lot of stuff this way. It’s particularly useful when you find yourself declaring a var just so you can initialize it with a loop. Any time you find yourself using var inside a function, consider whether this is better done with something else (map, filter, reduce, etc). Usually the answer is yes.

From whence would you return?

How about contains?

func contains<T: Equatable>(xs: [T], find: T) -> Bool {
  return xs.reduce(false) { found, x in found || find == x }

Hmmm…. It works, but it’s a poor choice in my opinion. reduce computes over the full list, so this is wasting a lot of effort checking values that aren’t needed. A good way to think about it is how you would expect this function to behave if the sequence were infinite. If you would never expect the function to return (like for sum), then reduce is proably a good fit. If you might expect it to return anyway (like for contains), then reduce is probably the wrong tool.

Another good way to think about it is whether you would include a return or break in your for-loop (or just use a while-loop). If so, your problem probably doesn’t match reduce well.

To make it more clear, let’s see how reduce can be implemented:

func reduce<T, U>(xs: [T], initial: U, combine: (U, T) -> U) -> U {
  var result = initial
  for x in xs {
    result = combine(result, x)
  return result

If your problem can be “reduced” to this, then reduce may be a good choice. If you need to stop the for-in loop, then you’ll probably want another tool like find.

So doctor, should I reduce?

I’ve given a bunch of examples here to show how powerful and flexible reduce is. I want to get your imagination going a little bit and give you some ideas on how to use this function to solve a wide range of problem. But just because you can write map in terms of reduce doesn’t mean you should.

Most of the time if your result is an array, mapping and filtering are the right tools. If you’re taking an array and deriving a single value, then reduce (or find) is often the best tool. But if you have a complex combination of mapping and filtering, then a single reduce may simplify that and also speed it up.

Reducing is a quite popular operation in functional languages. In fact, a lot of people (including your humble author) spend a lot of time trying to solve complicated problems with a single, elegant reduce. When I started this blog post, I got about half-way through implementing sorting that way when I realized I needed to stop. Reduce is the kind of function that makes people try to be clever, much like regular expressions. Resist the temptation, at least in your production code. Use the tools that make your code clear, not the tools that make you look smart.

But with that in mind, reduce your problems away!

  1. UPDATE: After some helpful discussion with @TacticalGrace and @pyrtsa, I’ve decided this description of fold right is too misleading. I usually favor simple-but-slightly-wrong explanations over more-correct-but-requires-lots-of-background-to-understand-and-is-still-slightly-wrong, but in this case, “goes backwards” really confuses how foldr is commonly used. We don’t have foldr in Swift, so I’m not going to dig into it here in a footnote; I’ll leave it to your exploration

  2. In this post, and in many of my posts, I will focus on Array implementations rather than the more general Sequence. That’s because Sequence (and Collection) can introduce a lot of complexity and obscure the key points being discussed. Some of their complexity is due to the nature of type constraints, and some is due to quirks about Swift that will likely improve. I’ve chosen to dodge all of that and focus on the most common and simplest case: Arrays. 

  3. If you’re familiar with Haskell, you may know that map is usually implemented as a foldr, while this is a foldl. That’s just for performance. In Haskell, list prepending (:) is much faster than appending (++). In Swift, array appending is the faster operation. So foldr is common in Haskell, but isn’t even in Swift stdlib. Keep this in mind when translating Haskell algorithms to Swift. Haskell often walks lists backwards for performance reasons. It’s more natural in Swift to walk them forward.