The Zipper is an idiom that uses the idea of "context" to the means of manipulating locations in a data structure. Zipper monad is a monad which implements the zipper for binary trees.
Sometimes you want to manipulate a location inside a data structure, rather than the data itself. For example, consider a simple binary tree type:
data Tree a = Fork (Tree a) (Tree a) | Leaf a
and a sample tree t:
t = Fork (Fork (Leaf 1) (Leaf 2)) (Fork (Leaf 3) (Leaf 4))
Each subtree of this tree occupies a certain location in the tree taken as a whole. The location consists of the subtree, along with the rest of the tree, which we think of the context of that subtree. For example, the context of
in the above tree is
Fork (Fork (Leaf 1) @) (Fork (Leaf 3) (Leaf 4))
where @ marks the spot that the subtree appears in. One way of expressing this context is as a path from the root of the tree to the required subtree. To reach our subtree, we needed to go down the left branch, and then down the right one. Note that the context is essentially a way of representing the tree, "missing out" a subtree (the subtree we are interested in).
We can represent a context as follows:
data Cxt a = Top | L (Cxt a) (Tree a) | R (Tree a) (Cxt a)
L c t represents the left part of a branch of which the right part was
t and whose parent had context
R constructor is similar.
Top represents the top of a tree. (Note that in the original paper, Huet dealt with B-trees (ones where nodes have arbitrary numbers of branches), so lists are used instead of the (Tree a) parameters.)
Using this datatype, we can rewrite the sample context above in proper Haskell:
R (Leaf 1) (L Top (Fork (Leaf 3) (Leaf 4)))
Note that the context is actually written by giving the path from the subtree to the root (rather than the other way round).
Now we can define a tree location:
type Loc a = (Tree a, Cxt a)
and some useful functions for manipulating locations in a tree:
left :: Loc a -> Loc a left (Fork l r, c) = (l, L c r) right :: Loc a -> Loc a right (Fork l r, c) = (r, R l c) up :: Loc a -> Loc a up (t, L c r) = (Fork t r, c) up (t, R l c) = (Fork l t, c) top :: Tree a -> Loc a top t = (t, Top) modify :: Loc a -> (Tree a -> Tree a) -> Loc a modify (t, c) f = (f t, c)
It is instructive to look at an example of how a location gets transformed as we move from root to leaf. Recall our sample tree t. Let's name some of the relevant subtrees for brevity:
t = let tl = Fork (Leaf 1) (Leaf 2) tr = Fork (Leaf 3) (Leaf 4) in Fork tl tr
(left . right . top) t = (left . right) (t, Top) = left (tr, R tl Top) = (Leaf 3, L (R tl Top) (Leaf 4))
There's a principled way to get the necessary types for contexts and the context filling functions, namely by differentiating the data structure. Some relevant papers.