[[!toc]]
##List Zippers##
Say you've got some moderately-complex function for searching through a list, for example:
let find_nth (test : 'a -> bool) (n : int) (lst : 'a list) : (int * 'a) ->
let rec helper (position : int) n lst =
match lst with
| [] -> failwith "not found"
| x :: xs when test x -> (if n = 1
then (position, x)
else helper (position + 1) (n - 1) xs
)
| x :: xs -> helper (position + 1) n xs
in helper 0 n lst;;
This searches for the `n`th element of a list that satisfies the predicate `test`, and returns a pair containing the position of that element, and the element itself. Good. But now what if you wanted to retrieve a different kind of information, such as the `n`th element matching `test`, together with its preceding and succeeding elements? In a real situation, you'd want to develop some good strategy for reporting when the target element doesn't have a predecessor and successor; but we'll just simplify here and report them as having some default value:
let find_nth' (test : 'a -> bool) (n : int) (lst : 'a list) (default : 'a) : ('a * 'a * 'a) ->
let rec helper (predecessor : 'a) n lst =
match lst with
| [] -> failwith "not found"
| x :: xs when test x -> (if n = 1
then (predecessor, x, match xs with [] -> default | y::ys -> y)
else helper x (n - 1) xs
)
| x :: xs -> helper x n xs
in helper default n lst;;
This duplicates a lot of the structure of `find_nth`; it just has enough different code to retrieve different information when the matching element is found. But now what if you wanted to retrieve yet a different kind of information...?
Ideally, there should be some way to factor out the code to find the target element---the `n`th element of the list satisfying the predicate `test`---from the code that retrieves the information you want once the target is found. We might build upon the initial `find_nth` function, since that returns the *position* of the matching element. We could hand that result off to some other function that's designed to retrieve information of a specific sort surrounding that position. But suppose our list has millions of elements, and the target element is at position 600512. The search function will already have traversed 600512 elements of the list looking for the target, then the retrieval function would have to *start again from the beginning* and traverse those same 600512 elements again. It could go a bit faster, since it doesn't have to check each element against `test` as it traverses. It already knows how far it has to travel. But still, this should seem a bit wasteful.
Here's an idea. What if we had some way of representing a list as "broken" at a specific point. For example, if our base list is:
[10; 20; 30; 40; 50; 60; 70; 80; 90]
we might imagine the list "broken" at position 3 like this (positions are numbered starting from 0):
40;
30; 50;
20; 60;
[10; 70;
80;
90]
Then if we move one step forward in the list, it would be "broken" at position 4:
50;
40; 60;
30; 70;
20; 80;
[10; 90]
If we had some convenient representation of these "broken" lists, then our search function could hand *that* off to the retrieval function, and the retrieval function could start right at the position where the list was broken, without having to start at the beginning and traverse many elements to get there. The retrieval function would also be able to inspect elements both forwards and backwards from the position where the list was "broken".
The kind of data structure we're looking for here is called a **list zipper**. To represent our first broken list, we'd use two lists: (1) containing the elements in the left branch, preceding the target element, *in the order reverse to their appearance in the base list*. (2) containing the target element and the rest of the list, in normal order. So:
40;
30; 50;
20; 60;
[10; 70;
80;
90]
would be represented as `([30; 20; 10], [40; 50; 60; 70; 80; 90])`. To move forward in the base list, we pop the head element `40` off of the head element of the second list in the zipper, and push it onto the first list, getting `([40; 30; 20; 10], [50; 60; 70; 80; 90])`. To move backwards again, we pop off of the first list, and push it onto the second. To reconstruct the base list, we just "move backwards" until the first list is empty. (This is supposed to evoke the image of zipping up a zipper; hence the data structure's name.)
We had some discussion in seminar of the right way to understand the "zipper" metaphor. I think it's best to think of the tab of the zipper being here:
t
a
b
40;
30; 50;
20; 60;
[10; 70;
80;
90]
And imagine that you're just seeing the left half of a real-zipper, rotated 60 degrees counter-clockwise. When the list is all "zipped up", we've "move backwards" to the state where the first element is targetted:
([], [10; 20; 30; 40; 50; 60; 70; 80; 90])
However you understand the "zipper" metaphor, this is a very handy data structure, and it will become even more handy when we translate it over to more complicated base structures, like trees. To help get a good conceptual grip on how to do that, it's useful to introduce a kind of symbolism for talking about zippers. This is just a metalanguage notation, for us theorists; we don't need our programs to interpret the notation. We'll use a specification like this:
[10; 20; 30; *; 50; 60; 70; 80; 90], * filled by 40
to represent a list zipper where the break is at position 3, and the element occupying that position is 40. For a list zipper, this is implemented using the pairs-of-lists structure described above.
##Tree Zippers##
Now how could we translate a zipper-like structure over to trees? What we're aiming for is a way to keep track of where we are in a tree, in the same way that the "broken" lists let us keep track of where we are in the base list.
It's important to set some ground rules for what will follow. If you don't understand these ground rules you will get confused. First off, for many uses of trees one wants some of the nodes or leafs in the tree to be *labeled* with additional information. It's important not to conflate the label with the node itself. Numerically one and the same piece of information---for example, the same `int`---could label two nodes of the tree without those nodes thereby being identical, as here:
root
/ \
/ \
/ \ label 1
/ \
label 1 label 2
The leftmost leaf and the rightmost leaf have the same label; but they are different leafs. The leftmost leaf has a sibling leaf with the label 2; the rightmost leaf has no siblings that are leafs. Sometimes when one is diagramming trees, one will annotate the nodes with the labels, as above. Other times, when one is diagramming trees, one will instead want to annotate the nodes with tags to make it easier to refer to particular parts of the tree. So for instance, I could diagram the same tree as above as:
1
/ \
2 \
/ \ 5
/ \
3 4
Here I haven't drawn what the labels are. The leftmost leaf, the node tagged "3" in this diagram, doesn't have the label `3`. It has the label 1, as we said before. I just haven't put that into the diagram. The node tagged "2" doesn't have the label `2`. It doesn't have any label. The tree in this example only has information labeling its leafs, not any of its inner nodes. The identity of its inner nodes is exhausted by their position in the tree.
That is a second thing to note. In what follows, we'll only be working with *leaf-labeled* trees. In some uses of trees, one also wants labels on inner nodes. But we won't be discussing any such trees now. Our trees only have labels on their leafs. The diagrams below will tag all of the nodes, as in the second diagram above, and won't display what the leafs' labels are.
Final introductory comment: in particular applications, you may only need to work with binary trees---trees where internal nodes always have exactly two subtrees. That is what we'll work with in the homework, for example. But to get the guiding idea of how tree zippers work, it's helpful first to think about trees that permit nodes to have many subtrees. So that's how we'll start.
Suppose we have the following tree:
9200
/ | \
/ | \
/ | \
/ | \
/ | \
500 920 950
/ | \ / | \ / | \
20 50 80 91 92 93 94 95 96
1 2 3 4 5 6 7 8 9
This is a leaf-labeled tree whose labels aren't displayed. The `9200` and so on are tags to make it easier for us to refer to particular parts of the tree.
Suppose we want to represent that we're *at* the node marked `50`. We might use the following metalanguage notation to specify this:
{parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
This is modeled on the notation suggested above for list zippers. Here `subtree 20` refers to the whole subtree rooted at node `20`:
20
/ | \
1 2 3
Similarly for `subtree 50` and `subtree 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:
{parent = ...; siblings = [*; subtree 920; subtree 950]}, * filled by subtree 500
And the parent of that targetted subtree should intuitively be a tree targetted on `node 9200`:
{parent = None; siblings = [*]}, * filled by tree 9200
This tree has no parents because it's the root of the base tree. Fully spelled out, then, our tree targetted on `node 50` would be:
{
parent = {
parent = {
parent = None;
siblings = [*]
}, * filled by tree 9200;
siblings = [*; subtree 920; subtree 950]
}, * filled by subtree 500;
siblings = [subtree 20; *; subtree 80]
}, * filled by subtree 50
In fact, there's some redundancy in this structure, at the points where we have `* filled by tree 9200` and `* filled by subtree 500`. Since node 9200 doesn't have any label attached to it, the subtree rooted in it is determined by the rest of this structure; and so too with `subtree 500`. So we could really work with:
{
parent = {
parent = {
parent = None;
siblings = [*]
},
siblings = [*; subtree 920; subtree 950]
},
siblings = [subtree 20; *; subtree 80]
}, * filled by subtree 50
We still do need to keep track of what fills the outermost targetted position---`* filled by subtree 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
For simplicity, I'll continue to use the abbreviated form:
{parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
But that should be understood as standing for the more fully-spelled-out structure. Structures of this sort are called **tree zippers**. They should already seem intuitively similar to list zippers, at least in what we're using them to represent. I think it may also be helpful to call them **targetted trees**, though, and so will be switching back and forth between these different terms.
Moving left in our targetted tree that's targetted on `node 50` would be a matter of shifting the `*` leftwards:
{parent = ...; siblings = [*; subtree 50; subtree 80]}, * filled by subtree 20
and similarly for moving right. If the sibling list is implemented as a list zipper, you should already know how to do that. If one were designing a tree zipper for a more restricted kind of tree, however, such as a binary tree, one would probably not represent siblings with a list zipper, but with something more special-purpose and economical.
Moving downward in the tree would be a matter of constructing a tree targetted on some child of `node 20`, with the first part of the targetted tree above as its parent:
{
parent = {parent = ...; siblings = [*; subtree 50; subtree 80]};
siblings = [*; leaf 2; leaf 3]
}, * filled by leaf 1
How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root node---let's call it `20'`---and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
node 20'
/ | \
/ | \
leaf 1 leaf 2 leaf 3
We'll call this new untargetted tree `subtree 20'`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `subtree 20'` being the subtree that fills that parent's target position `*`:
{
parent = ...;
siblings = [*; subtree 50; subtree 80]
}, * filled by subtree 20'
Or, spelling that structure out fully:
{
parent = {
parent = {
parent = None;
siblings = [*]
},
siblings = [*; subtree 920; subtree 950]
},
siblings = [*; subtree 50; subtree 80]
}, * filled by subtree 20'
Moving upwards yet again would get us:
{
parent = {
parent = None;
siblings = [*]
},
siblings = [*; subtree 920; subtree 950]
}, * filled by subtree 500'
where `subtree 500'` refers to a tree built from a root node whose children are given by the list `[*; subtree 50; subtree 80]`, with `subtree 20'` inserted into the `*` position. Moving upwards yet again would get us:
{
parent = None;
siblings = [*]
}, * filled by tree 9200'
where the targetted element is the root of our base tree. Like the "moving backward" operation for the list zipper, this "moving upward" operation is supposed to be reminiscent of closing a zipper, and that's why these data structures are called zippers.
We haven't given you a real implementation of the tree zipper, but only a suggestive notation. We have however told you enough that you should be able to implement it yourself. Or if you're lazy, you can read:
* [[!wikipedia Zipper (data structure)]]
* Huet, Gerard. ["Functional Pearl: The Zipper"](http://www.st.cs.uni-sb.de/edu/seminare/2005/advanced-fp/docs/huet-zipper.pdf) Journal of Functional Programming 7 (5): 549-554, September 1997.
* As always, [Oleg](http://okmij.org/ftp/continuations/Continuations.html#zipper) takes this a few steps deeper.