Wednesday, 10 December 2014


Our story today begins with the rather unmotivated type

eval :: Functor f => f (f a -> a) -> f a

Lets construct a value of the type,
eval f = xs where xs = fmap ($ xs) f

Plugging the list functor in, we have:

eval :: [[a] -> a] -> [a]

It becomes a little clearer from looking at the function that what we're doing is creating a list where each value in the list is a function of the list itself. The base case would be the constant function.
Creating a value of the argument to eval we have something like:

arg = [\list -> list !! 1, const 3, \list -> list !! 0 + list !! 1 + list !! 5const 2const 5const 4]

If we eval arg , it provides arg as the argument to each index in arg . If the graph isn't acyclic we're going to <<loop>>; this is very reminiscent of a spreadsheet.

Thus eval provides a method of traversal of a directed acyclic graph when the functor is a list. Among the many algorithms related to graph traversal I'm going to pick topological sort.
So each element in the list has a list of dependencies.

type AdjList = [[[Int]] -> [Int]]

topSort :: AdjList -> [Int]
topSort a = map fst res
     indices = [0..]
     res = sortBy comparator $ zip indices (eval a)
      comparator x y = compare (length (snd x)) (length (snd y))

We evaluate the dependencies and tag them with their indices. Then we sort them by the number of dependencies. The algorithm works because for any two nodes u and v, if there is a directed path from u to v, the number of dependencies associated with u is strictly greater than that of v.

What remains is to construct sample adjacency lists for which we will need to construct a list of dependencies:

ref = flip (!!)

deps l = foldl f (const []) l
      f g e = (++) <$> g <*> fmap (e:) (ref e)

adj1 = [deps [1, 2, 3], const [], deps [1, 3], const []]

Full code at

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