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.

Table of Contents

• Motivation
• Applicative Map?
• Combining Monoidal Operations
• Implementation

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 Ts by A thereby I can process all the Ts 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[]>>

(haskell)

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
          BATs[b][a].add Ts
  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 sequenceing 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

(Haskell)

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))
      )
    );
}