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# Inverting Map of Map in functional way

This post is about implementing a function for inverting Map of Maps, that is,

(typescript)

``````Map<A, Map<B, T>>
// to
Map<B, Map<A, T>>
``````

I’ll be using Haskell and Typescript throughout the post and show my best attempt in the end.

## Motivation

Recently I got to deal with lots of instances of a type `T`. Since there were so many of them, I had to group them by some criterion for performance reasons. For example, `T` uses resource `A` so I group `T`s by `A` thereby I can process all the `T`s that share the same `A` at once when it is allocated.

In my case there were two criteria, `A` and `B`, that I had to take into account simultaneously so naturaly I got to build a data structure of

(typescript)

``````Map<A, Map<B, T[]>>
``````

``````Map A (Map B [T])
``````

The problem was that in some cases it needed to be expressed in the form of `Map<B, Map<A, T[]>>` which is alteration of switching the outter Map’s Key and inner Map’s Key.

Of course I could just implement it imperatively something like this …

(pseudo imperative code)

``````// boring imperative code …
func invertMap(ABTs: Map<A, Map<B, T[]>>):
BATs = new Map<B, Map<A, T[]>>
for (a, BTs) of ABTs:
for (b, Ts) of BTs:
if BATs has no key b then
BATs[b] = new Map<A, T[]>
BATs[b][a] = Ts
else
if BATs[b] has no key a then
BATs[b][a] = Ts
else
return BATs
``````

But I thought I could do better. They were the algebraically same data structure therefore there must have been a natural way of transforming into each other. I only needed to find them.

## Applicative Map?

At first I thought about conforming `Map<B, _>` to Applicative then just `sequence`ing it. It was possible because Map is Traversable. But soon I found Map couldn’t conform to Applicative because there is simply no way to implement `pure` (or `of`, if you will) nor `ap` (or `<*>`, `liftA`, if you will). I mean how are you going to construct a Map of functions that satisfies Identity law?!

So I gave up on Applicative Map.

## Combining Monoidal Operations

I got thinking that it was evident in the imperative version of implementation above that the whole process is just combination of Monoidal operations: creating empty Map, concatenating list of Ts and ultimately combining all to define monoidal operation on `Map<B, Map<A, T>>`.

This observation led me to this functional implementation.

## Implementation

``````invertMap :: (Monoid t, Ord a, Ord b) => Map a (Map b t) -> Map b (Map a t)
invertMap = foldr (unionWith mappend) empty . mapWithKey (fmap . singleton)
``````

(Typescript, fp-ts)

``````
export function invertMap<KA, KB, T>(
kaOrd: Eq<KA>,
kbOrd: Eq<KB>,
monT: Monoid<T>
) {
const monAT = map.getMonoid(kaOrd, monT);
const monBAT = map.getMonoid(kbOrd, monAT);
return (mm: Map<KA, Map<KB, T>>) =>
getFoldableWithIndex(kaOrd).foldMapWithIndex(monBAT)(mm, (a, bt) =>
pipe(
bt,
map.map((t) => new Map<KA, T>().set(a, t))
)
);
}
``````