Reducing Map to Reduce OR All You Need is Fold

"That would reduce these bloody days again"
-- Shakespeare, Richard III
Act V, Scene V, Line 36

Intro

Have you heard the tale of the madman who came running into the town late at night, screaming that all you need is fold?

In Graham Hutton's paper "A tutorial on the universality and expressiveness of fold" it is shown that we can reduce recursive functions into a definition using fold.  Let's take a look reducing map to a fold.

(O-the tale of the madman is about me and not Dr. Hutton.)

Map

Map goes by a few different names, to those who speak Clojure, Haskell, or Underscore.js it goes by the name Map, to those which speak C# it goes by the name Select, those who speak C++ call it Transform, ... we could go on like this for a bit.

The point is that idea of the higher order function Map goes by different names in different language but performs the same function of taking a collection and applying a function against each member and placing it into the resulting collection.

The easiest example that actual does something is to apply an increment function with a Map.  Looking above we see that when we Map the increment function against a collection with the members of 1, 2, 3, 4 we get a resulting collection with the members of 2, 3, 4, 5.

In Clojure and C# it would look like this:

Fold

Fold goes by a few different names too, to those that speak Clojure or Underscore.js it goes by the name Reduce, to those which speak C# it is known as Aggregate, in Haskell it is known as Fold, in C++ it goes by the name Accumulate, ... we could go on like this for a while.  Again the point is that these names are all for the same idea of the higher order function Fold.

A Fold takes a collection and applies one function against each member of the collection and an accumulated result (also known as a memorize).

A very simple example of Fold would be summing.

We see in the image above that we are taking the collection 1, 2, 3, 4 and summing it up to the value of 10.  We do this by taking member 1 and the seed value of 0 and adding them together to get 1.  Next, we take the accumulated value of 1 and add the next member 2 to it giving us 3.  Then, we take the accumulated value of 3 and the member 3 add them and get 6.  Last, we take the accumulated value of 6 and the member of 4 giving us the final result of 10.

In Clojure and C# this would look like the following:

Mapping Map to Fold

... and birds go tweet, what is the point?  Glad you asked, we see with both Map and Fold some similarities.

1. applying a function against each member of a collection
2. building a resulting collection

Let's look at the first Map example again.  We see that we applying a function against each member of a collection and are accumulating a resulting collection, this sounds very similar to Fold.

In fact I would say that if we had an empty collection in which we accumulated the results of applying the function against the members of the collection we would have the same functionality.  That was a lot of words lets have some code.

In Clojure and C# we would have the following:

We see that both Map and the Fold give the exact same result.  In both the Clojure and C# code we see that we need an initial empty collection to store the resulting collection in and place the result of applying the function against a member into the resulting collection (using conj in Clojure and Add in C#).  For the C# code since we are using a List to accumulated the resulting collection instead of an array we need to reshape the resulting collection using the ToArray method.

By looking at this simple example we see that we can reduce Map to a Fold and thus all we need is Fold.  :)

The city of Mapcat

"I have forgot the map."
-- Shakespeare, Henry IV Part I
Act III, Scene I, Line 5.1

Mapcat is an interesting higher order function which is not talked about much.

"You've never danced like this before

We don't talk about it"

-- Milky Chance,

Stolen Dance

The simplest case of Mapcat is when you have a collection of collection and you want to map over the collection of collection yielding a single level of collection.

In the image above we see that we have a collection of two collections (top and bottom) and a function which maps and cats (mapper catter) when these are Mapcatted that yield a result which is a single collection.  The size of the result from the Mapcat is the size of the collection's collections, in the image above we see that we have 4 members of the first collection and 2 members in the second collection giving us the resulting collection with 6 members.

The simplest example of Mapcat would be using the identity function (a function which just returns whatever is given to it).

We see that collection which contains a collection we 1, 2, 3, and 4 and another collection with 20 and 10 when Mapcatted with the Identity function will yield a resulting collection with 1, 2, 3, 4, 20, and 10.



We see in C# we can use the SelectMany to perform this simple Mapcat, while Clojure has a function called mapcat.

This is nice but how about a simple example which actual does something?

How about incrementing the values?



We see that in both C# and Clojure the lambda in the Mapcat is given a collection, so in both cases we need to have a function which can work with a collection.  We have a choice in both C# and Clojure we can either have a Map in the lambda or we can chain the result by removing a level with the identity function follow by a simple Map with an incrementing lambda function.  In Clojure we can do chaining using the double arrow macro (thread last) ->> or just nest the calls.

On Zipping

"With those legions
Which I have spoke of, whereunto your levy
Must be supplyant: the words of your commission
Will tie you to the numbers and the time
Of their despatch."

-- Shakespeare, Cymbeline

Act III, Scene VIII, Lines 12.2 - 16.1

Zipping is a very simple concept from higher order functions which does not get talked about much.

With a zip you have a number of collection which you wish to apply a function against each matching member in order to produce a single "zipped" collection.

In general:

GIVEN a function f

FOREACH x_i in X, y_i in Y, ...
  r_i = f(x_i, y_i, ..)

GIVING r_i in R

The function f is the "zipper", while X, Y, ... are the collections being zipped with R being the zipped result.

A simple example with 2 collections:

Let's use Add as the "zipper", [1, 2, 3] as the first collection, and [10, 20, 30] as the second collection.

We see that this gives us the resulting collection of [11, 22, 33].

What should happen if the first or second collection has a different number of elements?

OR

We'll look at what C#, Clojure, and Haskell do in these cases.



We see that they each have a resulting collection as large as the smallest input collection and simply just ignore the rest of the input in the larger collection.  There you have it a very simple introduction to Zip.