Directed Acyclic Graphs

Posted on February 3, 2017
Directed Acyclic Graphs - HackMD

Representing graph

There are a few ways to represent a graph:

  • as a list of Nodes and a list of edges
  • as a list of neighbors

each representation is optimized for a runtime behaviour (cf Kormen) but
it’s useful to abstract representation to express operations. A
convenient way to abstract over those representations is shown in Fun
with type

But what if we could abuse haskell somehow to represent graph not with
some external description but with haskell itself ?

Breaking referential transparency

We want to write graph that way

graph1 :: Tree IntTreeF
graph1 = iNode (iNode (iLeaf 2) a) b
           where a = iNode (iNode (iLeaf 2)(iLeaf 2)) (iLeaf 2)
                 b = a

instead of some variation of

-- graph1' = mkDag (Node (Node (Leaf 2) (Ref 0)) (Ref 0)) [(0, Node (Node (Leaf 2) (Leaf 2)) (Leaf 2))]

But normally, in Haskell, there should be no way to distinguish graph1 from graph2 below

graph2 :: Tree IntTreeF
graph2 =  iNode (iNode (iLeaf 2) a) b
           where a = iNode (iNode (iLeaf 2) (iLeaf 2)) (iLeaf 2)
                 b = iNode (iNode (iLeaf 2) (iLeaf 2)) (iLeaf 2)

But we can actually abuse haskell by going to IO and relying on some
implementation of GHC to observe the sharing using Type-safe observable sharing in Haskell a technique implemented and extended in data-reify

With this we can produce two different output for two values which
should have been be undistinguishable.

graph1 graph2
graph1 graph2

basically the second graph is maximal sharing of subnodes. because we
only ever use names like a and b once, they are irrelevant. whereas in
the first case, this bottom leaf 2 is not really any leaf 2, it is the
leaf which comes from a which should also be equal to b as a graph.
to make sure they are equal, the equivalent relation a = b gets its own
private copy which is not shared with other subgraphs.

using this we can directly represent graphs with haskell which is
convienent and more visual than labeling nodes. indeed compare the code,
visual, with the following ‘normal’ representation of first and second


  • traversals (control - cps, mendler, launchbury coroutining fold)
  • names (polymorphism)
  • regions (applicative batching, compdata)