types - How do you represent a graph in Haskell?

Question

It's easy enough to represent a tree or list in haskell using algebraic data types. But how would you go about typographically representing a graph? It seems that you need to have pointers. I'm guessing you could have something like

type Nodetag = String
type Neighbours = [Nodetag]
data Node a = Node a Nodetag Neighbours

And that would be workable. However it feels a bit decoupled; The links between different nodes in the structure don't really "feel" as solid as the links between the current previous and next elements in a list, or the parents and children of a node in a tree. I have a hunch that doing algebraic manipulations on the graph as I defined it would be somewhat hindered by the level of indirection introduced through the tag system.

It is primarily this feeling of doubt and perception of inelegance that causes me to ask this question. Is there a better/more mathematically elegant way of defining graphs in Haskell? Or have I stumbled upon something inherently hard/fundamental? Recursive data structures are sweet, but this seems to be something else. A self referential data structure in a different sense to how trees and lists are self referential. It's like lists and trees are self referential at the type level, but graphs are self referential at the value level.

So what's really going on?

Answer 1

In shang's answer you can see how to represent a graph using laziness. The problem with these representations is that they are very difficult to change. The knot-tying trick is useful only if you're going to build a graph once, and afterward it never changes.

In practice, should I actually want to do something with my graph, I use the more pedestrian representations:

• Edge list
• Adjacency list
• Give a unique label to each node, use the label instead of a pointer, and keep a finite map from labels to nodes

If you're going to be changing or editing the graph frequently, I recommend using a representation based on Huet's zipper. This is the representation used internally in GHC for control-flow graphs. You can read about it here:

• An Applicative Control-Flow Graph based on Huet's Zipper

• Hoopl: A Modular, Reusable Library for Dataflow Analysis and Transformation