# Sorting with Style

Merge sort is a famous comparison-based sorting algorithm that starts by first recursively dividing a collection of orderable elements into smaller subcollections, and then finishes by recursively sorting and merging the smaller subcollections together to reconstruct the (now sorted) original.

A clear implementation of mergesort should by definition be as faithful to that high-level description as possible. We can get pretty close to that using the whole recursion schemes business that I’ve talked about in the past. Near the end of that article I briefly mentioned the idea of implementing mergesort via a hylomorphism, and here I just want to elaborate on that a little.

Start with a collection of orderable elements. We can divide the collection into a bunch of smaller collections by using a binary tree:

{-# LANGUAGE DeriveFunctor #-}

import Data.Functor.Foldable (hylo)
import Data.List.Ordered (merge)

data Tree a r =
Empty
| Leaf a
| Node r r
deriving Functor

The idea is that each node in the tree holds two subtrees, each of which contains half of the remaining elements. We can build a tree like this from a collection - say, a basic Haskell list. The following unfolder function defines what part of a tree to build for any corresponding part of a list:

unfolder []  = Empty
unfolder [x] = Leaf x
unfolder xs  = Node l r where
(l, r) = splitAt (length xs div 2) xs

On the other hand, we can also collapse an existing tree back into a list. The following folder function defines how to collapse any given part of a tree into the corresponding part of a list; again we just pattern match on whatever part of the tree we’re looking at, and construct the complementary list:

folder Empty      = []
folder (Leaf x)   = [x]
folder (Node l r) = merge l r

Now to sort a list we can just glue these instructions together using a hylomorphism:

mergesort :: Ord a => [a] -> [a]
mergesort = hylo folder unfolder

And it works just like you’d expect:

> mergesort [1,10,3,4,5]
[1,3,4,5,10]
> mergesort "aloha"
"aahlo"
> mergesort [True, False, False, True, False]
[False, False, False, True, True]

Pretty concise!

The code is eminently clean and faithful to the high-level algorithm description: first recursively divide a collection into smaller subcollections - via a binary tree and unfolder - and then recursively sort and merge the subcollections to reconstruct the (now sorted) original one - via folder.

A version of this post originally appeared on the Fugue blog.