Finger Trees are definitely the coolest data structure I was never taught in school. The gist of Finger Trees is that they represent sequences of elements where the elements also have a measurable 'descriptor' of some kind. If that sounds vague it's because it is! The generality here is what allows Finger Trees to solve so many different types of problems, but it does require a few examples and explanations to understand. In this post we'll talk about how the trees work at a high level, then we'll use them to build a random-access array-like structure with reasonable performance characteristics.

This data structure stands on the shoulders of giants, it uses a structure called a `Monoid`

at its core.

## Monoids

If you're entirely unfamiliar with the concept of monoids, or just need a refresher, it would be a good idea to get a solid grounding there first; here's a good place to start.

Monoids are incredibly useful; and the more I learn about Category Theory the more applications I find for monoidal structures. Once you start to think in monoids you start to realize how many things you once thought were unique and interesting problems are actually just a monoid and a fold away from some other well-solved problem. We're going to start off by introducing a new tool (i.e. data structure) which employs monoids to do amazing things! Enter Finger Trees! Finger Trees are an adaptable purely functional data structure; they're actually an extremely general structure which makes it a bit difficult to explain without a concrete use case. This is because they utilize a Monoid in the foundation of the data structure, and the Monoid you choose can drastically affect how the structure behaves. Here's a glance at the sort of things you could do by choosing different Monoids:

- Random access/sequence slicing using
`Sum`

: see Data.Sequence; we'll explore this one just below! - Heap using
`Max/Min`

: see Data.PriorityQueue.FingerTree - Ordered Sequence slicing using Last: see the section on Ordered Sequences
- Interval Searching using a custom interval expansion Monoid: see Data.IntervalMap.FingerTree
- Text slicing and dicing using a product of
`Sum`

s: see Yi.Rope - Performant merge sort using a custom merge monoid: blog post coming eventually!
- Many more! Just use your imagination!

How does it all work? Let's learn how to build a simple random-access <air-quotes> Array </air-quotes> using a Finger Tree so we can get a sense of things

## Random Access Array using a Finger Trees

Let's implement a simple random access list using a Finger Tree! After a quick glance through the Data.FingerTree Docs it's a bit tough to tell where we might start! The workhorse of the Finger Tree library is the `split`

function:

Yikes, let's break this down:

`Measured v a`

: Measured is a simple typeclass which given an`a`

can convert it into some monoid`v`

`(v -> Bool)`

: This is our search predicate,`split`

will use it to split a sequence into two smaller subsequences: The longest subsequence such that running the predicate on the measure of this subsequence`False`

, and the everything that's left-over.`FingerTree v a`

: This is the tree we want to split, with a monoidal measure`v`

and elements of type`a`

.`(FingerTree v a, FingerTree v a)`

: The two (possibly empty) subsequences, the first is before the split point the second contains the inflection point of our predicate and everything past it.

That's all great, but how can we actually use it to solve our problem? What does splitting up a sequence actually have to do with indexing into a list? Finger Trees get their performance characterics by searching through subtrees using a strategy very similar to a binary search, they run the predicate on cached "measures" of subtrees recursively honing in on the **inflection point** where the predicate flips from `False`

to `True`

. So what we need to do is find some pairing of a monoid and a predicate on that monoid which finds the place in the sequence we're looking for. Getting the first or last element of a Finger Tree is a simple `O(1)`

operation, so if we can split the list either directly *before* or directly *after* the index we're looking for, then we're pretty much done!

Building a predicate for this is pretty simple, we just need to be able to determine whether the index we're looking for is within some prefix of our total sequence, which phrased simply is just: `length sequence > index`

; we can use this predicate to recursively hone in on the point where adding a single element alters the predicate's result from false to true, and we've found our index! The predicate runs on the measure of the values, which must be a monoid; so we need to represent the length of our sequence as some monoid, the combination of the monoidal measure of two sequences must also match the measure of the combination of the sequences themselves! Luckily for us the length of the combination of two lists is just the sum of the lengths! This gives us the hint that we can use the `Sum`

Monoid as our measure!

We're so close now, let's write some code to make it happen.

```
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
import Data.FingerTree
import Data.Monoid
-- We need to wrap our primitive value type in a newtype;
-- This allows us to store ANY value in the sequence and helps us avoid
-- some trouble with functional dependencies and orphan instances.
newtype Size a = Size
{ getSize :: a
} deriving (Show, Eq)
-- Measured is the typeclass we implement to tell the FingerTree how to measure
-- our values into a monoid. In our case every individual element is simply of length '1'
instance Measured (Sum Int) (Size a) where
measure _ = Sum 1
-- We wrap our values in the 'Size'i wrapper and build a Finger Tree
alphabet :: FingerTree (Sum Int) (Size Char)
alphabet = fromList (fmap Size "abcdefghijklmnopqrstuvwxyz")
-- Get a given index from the tree if it exists
atIndex :: Int -> FingerTree (Sum Int) (Size a) -> Maybe a
atIndex n t =
case viewl . snd $ split (> Sum n) t of
Size c :< _ -> Just c
_ -> Nothing
```

Hopefully the first bits are pretty self explanatory, we set up our datatypes so the tree knows how to measure our elements, and it already knows how to combine measures via Sum's Monoid instance. Lastly in `atIndex`

we tell the tree to split open at the point where the length of the measured subsequence would surpass the index we've provided. Then we simply check if there's an element to the right of that split. This operation doesn't quite get us the `O(1)`

time complexity we know and love from traditional arrays, but for an immutable, general data structure which we could build ourselves without ANY special compiler support, getting logarithmic performance isn't too bad. In fact the actual performance is `O(log(min(i,n-i)))`

where `i`

is the index we wish to access and `n`

is the length of the sequence. If we're often accessing the first or last elements then we're down to pretty much constant time!

There we go! We've used 'Sum' as a measure within a finger tree to get efficient indexing into a sequence! We can also notice that the length of the whole sequence is computed in `O(1)`

if we use `length = measure`

; and that we can concat two sequences relatively efficiently using `(><)`

; listed in `Data.FingerTree`

as time complexity `O(log(min(n1, n2)))`

where n1 and n2 are the length of each sequence respectively.

`Sum`

is probably the simplest monoid we can use; take a minute to think about how other monoids you know of might behave; the majority of monoids will create SOME sort of useful structure when used with a Finger Tree!

Hopefully you learned something 🤞! If you did, please consider checking out my book: It teaches the principles of using optics in Haskell and other functional programming languages and takes you all the way from an beginner to wizard in all types of optics! You can get it here. Every sale helps me justify more time writing blog posts like this one and helps me to continue writing educational functional programming content. Cheers!