Posted on February 20, 2020
Part 9 of a 10-part series on Breadth-First Traversals

This post will be quite light on details: I’m trying to gather up all of the material in this series to be a chapter in my Master’s thesis, so I’m going to leave the heavy-duty explanations and theory for that. Once finished I will probably do a short write up on this blog.

That said, the reason I’m writing this post is that in writing my thesis I figured out a nice way to solve the problem I first wrote about in this post. I won’t restate it in its entirety, but basically we’re looking for a function with the following signature:

``bft :: Applicative f => (a -> f b) -> Tree a -> f (Tree b)``

Seasoned Haskellers will recognise it as a “traversal”. However, this shouldn’t be an ordinary traversal: that, after all, can be derived automatically by the compiler these days. Instead, the Applicative effects should be evaluated in breadth-first order. To put it another way, if we have a function which lists the elements of a tree in breadth-first order:

``bfs :: Tree a -> [a]``

Then we should have the following identity:

``bft (\x -> ([x], x)) t = (bfs t, t)``

Using the writer Applicative with the list monoid here as a way to talk about ordering of effects.

There are many solutions to the puzzle (see Gibbons 2015; or Easterly 2019, or any of the posts in this series), but I had found them mostly unsatisfying. They basically relied on enumerating the tree in breadth-first order, running the traversal on the intermediate list, and then rebuilding the tree. It has the correct time complexity and so on, but it would be nice to deforest the intermediate structure a little bit more.

Anyways, the function I finally managed to get is the following:

``````bft :: Applicative f => (a -> f b) -> Tree a -> f (Tree b)
bft f (x :& xs) = liftA2 (:&) (f x) (bftF f xs)

bftF :: Applicative f => (a -> f b) -> [Tree a] -> f [Tree b]
bftF t = fmap head . foldr (<*>) (pure []) . foldr f [pure ([]:)]
where
f (x :& xs) (q : qs) = liftA2 c (t x) q : foldr f (p qs) xs

p []     = [pure ([]:)]
p (x:xs) = fmap (([]:).) x : xs

c x k (xs : ks) = ((x :& xs) : y) : ys
where (y : ys) = k ks``````

The `Tree` is defined like so:

``data Tree a = a :& [Tree a]``

It has all the right properties (complexity, etc.), and if you stick tildes before every irrefutable pattern-match it is also maximally lazy.

As a bonus, here’s another small function I looked at for my thesis. It performs a topological sort of a graph.

``````type Graph a = a -> [a]

topoSort :: Ord a => Graph a -> [a] -> [a]
topoSort g = fst . foldr f ([], ∅)
where
f x (xs,s)
| x ∈ s = (xs,s)
| x ∉ s = first (x:) (foldr f (xs, {x} ∪ s) (g x)) ``````

Easterly, Noah. 2019. “Functions and Newtype Wrappers for Traversing Trees: Rampion/Tree-Traversals.” https://github.com/rampion/tree-traversals.