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Programming language: Haskell
License: BSD 3-clause "New" or "Revised" License
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Nested Map

Where the fundamental signature for map is k -> v, for a nested map it's [k] -> [v]. Or perhaps slightly more accurately, something like k -> k -> k -> ... -> [v].

One way of thinking about Map is as the tabulation or memorization of a function, or perhaps the transformation from 'time' to 'space'. In the first conception, we could think about NestedMap as the shape preserving transformation of the composition of functions; in the second interpretation, we could view NestedMap as a shape preserving transformation of structured time to structured space.

Alternately, we could just look it as nice implemetation of parametric sparse tensor.

For version 2.0, it would be nice to change [(k,v)] into a heterogenous list. For now, this can be poorly addressed using sum or existential types.

How is this at all useful?

Well, for example, it makes the implementation of an efficent markov chain trivial.

It's also useful for eliminating bulky conditions on transformations of complex data. By first transforming the data into a nested map representation via a list of projections, it's much easier write the logic for tranforming and operating on the data--since at any particular node in the nested map, you are guaranteed to not only be working with only the data described by the ancestors, but also all of the prescribed data.

Another example might be delineating highways from local roads given a data set of directions, after doing a topological sort.

It is very useful for any sort of number crunching on sparse matricies or tensors.

Why not just use Map [k] v?

Well, we could of course. However, we're often very interested in our context--the coordinates of where we live. While Map [k] v could make this theoretically possible, it's horrible inefficent, and would make it much harder to guarentee structural soundness.

For example, we could easily delete an interior node while leaving the descendents intact. That's just not possible with NestedMap.

Isn't this just a Trie?

Well, it kind of is. But in practice, tries seem to be focused on strings as keys. We want to be more general. I'm sure there is a better name.

Isn't this just a RoseTree?

Well, it kind of is. But the RoseTree is just RoseTree a, where as we are Tree k a or Forest k a. Of course, you could use a pair and define the Ord and Eq instance appropriately, but it's using lists underneath, which is a performance killer for large data sets.

If a Tree falls in a Forest, and no one is there, does it make a sound?

Every cat belongs to Schrödinger in the land of the lazy.

Implementation Notes

So, there are two interfaces: Tree and Forest.

Neither one is favored over the other. Use Tree if you need a top level value.

Theoretically, these mutually recursive shapes should be completely independent of each other, perhaps just exposed to the other via a typeclass. However, mutually recursive modules are very difficult to work with, even with GHC. Perhaps one day. In the mean time, the implementation doesn't matter so much and won't affect client code as long as the public Tree and Forest interfaces are used.