##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. For simplicity, we'll only implement this for binary trees---trees where internal nodes always have exactly two subtrees. But to get the guiding idea, it's helpful first to think about trees which permit nodes to have many subtrees. Suppose we had the following tree:
+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. In a particular application, you may only need to implement this for binary trees---trees where internal nodes always have exactly two subtrees. But to get the guiding idea, it's helpful first to think about trees that permit nodes to have many subtrees. Suppose we have the following tree:
1000
/ | \
20 50 80 91 92 93 94 95 96
1 2 3 4 5 6 7 8 9
-and we want to represent that we're at the node labeled `50`. We might use the following metalanguage notation to specify this:
+and we want to represent that we're at the node marked `50`. We might use the following metalanguage notation to specify this:
{parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
-This is modeled on the notation suggested above for list zippers. Here `node 20` refers not to a `int` label attached to that node, but rather to the whole subtree rooted at that node:
+This is modeled on the notation suggested above for list zippers. Here `node 20` refers not to a `int` label associated with that node, but rather to the whole subtree rooted at that node:
20
/ | \
siblings = [node 20; *; node 80]
}, * filled by node 50
-We're understanding the `20` here in `node 20` to just be a metalanguage tag to help us theorists keep track of which node we're referring to. We're supposing the tree structure itself doesn't associate any informative labelling information with those nodes. It only associates informative labels with the tree leafs. (We haven't represented any such labels in our diagreams.)
+We're understanding the `20` here in `node 20` to just be a metalanguage marker to help us theorists keep track of which node we're referring to. We're supposing the tree structure itself doesn't associate any informative labelling information with those nodes. It only associates informative labels with the tree leafs. (We haven't represented any such labels in our diagrams.)
We still do need to keep track of what fills the outermost targetted position---`* filled by node 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
{parent = ...; siblings = [*; node 50; node 80]}, * filled by node 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 represented siblings with a list zipper, but with something more special-purpose and economical.
+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:
node 20
/ | \
/ | \
- lead 1 leaf 2 leaf 3
+ leaf 1 leaf 2 leaf 3
-We'll call this new untargetted tree `node 20`. The result of moving upward from our tree targetted on `leaf 1` would be the outermost `parent` element of that targetted tree, with `node 20` being the subtree that occupies that parent's `*` position:
+We'll call this new untargetted tree `node 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 `node 20` being the subtree that fills that parent's target position `*`:
{
parent = ...;
* 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.
-[[Same-fringe using a tree zipper]]
+##Same-fringe using a tree zipper##
Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the targetted leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost position in a sibling list), more upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be targetted on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:
you won't move upwards at the same steps. Keep comparing "the next leafs" until they are different, or you exhaust the leafs of only one of the trees (then again the trees have different fringes), or you exhaust the leafs of both trees at the same time, without having found leafs with different labels. In this last case, the trees have the same fringe.
+If your trees are very big---say, millions of leaves---you can imagine how this would be quicker and more memory-efficient than traversing each tree to construct a list of its fringe, and then comparing the two lists so built to see if they're equal. For one thing, the zipper method can abort early if the fringes diverge early, without needing to traverse or build a list containing the rest of each tree's fringe.
+
+Let's sketch the implementation of this. We won't provide all the details for an implementation of the tree zipper, but we will sketch an interface for it.
+
+First, we define a type for leaf-labeled, binary trees:
+
+ type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
+
+Next, the interface for our tree zippers. We'll help ourselves to OCaml's **record types**. These are nothing more than tuples with a pretty interface. Instead of saying:
+
+ # type blah = Blah of (int * int * (char -> bool));;
+
+and then having to remember which element in the triple was which:
+
+ # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
+ Error: This expression has type int * (char -> bool) * int
+ but an expression was expected of type int * int * (char -> bool)
+ # (* damnit *)
+ # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
+ val b1 : blah = Blah (1, 2, <fun>)
+
+records let you attach descriptive labels to the components of the tuple:
+
+ # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
+ # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
+ val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
+ # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *)
+ val b3 : blah_record = {height = 1; weight = 3; char_tester = <fun>}
+
+These were the strategies to extract the components of an unlabeled tuple:
+
+ let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)
+
+ let (h, w, test) = b1;; (* works for arbitrary tuples *)
+
+ match b1 with
+ | (h, w, test) -> ...;; (* same as preceding *)
+
+Here is how you can extract the components of a labeled record:
+
+ let h = b2.height;; (* handy! *)
+
+ let {height = h; weight = w; char_tester = test} = b2
+ in (* go on to use h, w, and test ... *)
+
+ match test with
+ | {height = h; weight = w; char_tester = test} ->
+ (* go on to use h, w, and test ... *)
+
+Anyway, using record types, we might define the tree zipper interface like so:
+
+ type 'a starred_tree = Root | Starring_Left of 'a starred_pair | Starring_Right of 'a starred_pair
+ and 'a starred_pair = { parent : 'a starred_tree; sibling: 'a tree }
+ and 'a zipper = { tree : 'a starred_tree; filler: 'a tree };;
+
+ let rec move_botleft (z : 'a zipper) : 'a zipper =
+ (* returns z if the targetted node in z has no children *)
+ (* else returns move_botleft (zipper which results from moving down and left in z) *)
+
+<!--
+ let {tree; filler} = z
+ in match filler with
+ | Leaf _ -> z
+ | Node(left, right) ->
+ let zdown = {tree = Starring_Left {parent = tree; sibling = right}; filler = left}
+ in move_botleft zdown
+ ;;
+-->
+
+ let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
+ (* if it's possible to move right in z, returns Some (the result of doing so) *)
+ (* else if it's not possible to move any further up in z, returns None *)
+ (* else returns move_right_or_up (result of moving up in z) *)
+
+<!--
+ let {tree; filler} = z
+ in match tree with
+ | Starring_Left {parent; sibling = right} -> Some {tree = Starring_Right {parent; sibling = filler}; filler = right}
+ | Root -> None
+ | Starring_Right {parent; sibling = left} ->
+ let z' = {tree = parent; filler = Node(left, filler)}
+ in move_right_or_up z'
+ ;;
+-->
+
+The following function takes an 'a tree and returns an 'a zipper focused on its root:
+
+ let new_zipper (t : 'a tree) : 'a zipper =
+ {tree = Root; filler = t}
+ ;;
+
+Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:
+
+ let make_fringe_enumerator (t: 'a tree) =
+ (* create a zipper targetting the root of t *)
+ let zstart = new_zipper t
+ in let zbotleft = move_botleft zstart
+ (* create a refcell initially pointing to zbotleft *)
+ in let zcell = ref (Some zbotleft)
+ (* construct the next_leaf function *)
+ in let next_leaf () : 'a option =
+ match !zcell with
+ | None -> (* we've finished enumerating the fringe *)
+ None
+ | Some z -> (
+ (* extract label of currently-targetted leaf *)
+ let Leaf current = z.filler
+ (* update zcell to point to next leaf, if there is one *)
+ in let () = zcell := match move_right_or_up z with
+ | None -> None
+ | Some z' -> Some (move_botleft z')
+ (* return saved label *)
+ in Some current
+ )
+ (* return the next_leaf function *)
+ in next_leaf
+ ;;
+
+Here's an example of `make_fringe_enumerator` in action:
+
+ # let tree1 = Leaf 1;;
+ val tree1 : int tree = Leaf 1
+ # let next1 = make_fringe_enumerator tree1;;
+ val next1 : unit -> int option = <fun>
+ # next1 ();;
+ - : int option = Some 1
+ # next1 ();;
+ - : int option = None
+ # next1 ();;
+ - : int option = None
+ # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
+ val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
+ # let next2 = make_fringe_enumerator tree2;;
+ val next2 : unit -> int option = <fun>
+ # next2 ();;
+ - : int option = Some 1
+ # next2 ();;
+ - : int option = Some 2
+ # next2 ();;
+ - : int option = Some 3
+ # next2 ();;
+ - : int option = None
+ # next2 ();;
+ - : int option = None
+
+You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will keep returning `None`.
+
+Using these fringe enumerators, we can write our `same_fringe` function like this:
+
+ let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
+ let next1 = make_fringe_enumerator t1
+ in let next2 = make_fringe_enumerator t2
+ in let rec loop () : bool =
+ match next1 (), next2 () with
+ | Some a, Some b when a = b -> loop ()
+ | None, None -> true
+ | _ -> false
+ in loop ()
+ ;;
+
+The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itself---either because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.
+
+The technique illustrated here with our fringe enumerators is a powerful and important one. It's an example of what's sometimes called **cooperative threading**. A "thread" is a subprogram that the main computation spawns off. Threads are called "cooperative" when the code of the main computation and the thread fixes when control passes back and forth between them. (When the code doesn't control this---for example, it's determined by the operating system or the hardware in ways that the programmer can't predict---that's called "preemptive threading.") Cooperative threads are also sometimes called *coroutines* or *generators*.
+
+With cooperative threads, one typically yields control to the thread, and then back again to the main program, multiple times. Here's the pattern in which that happens in our `same_fringe` function:
+
+ main program next1 thread next2 thread
+ ------------ ------------ ------------
+ start next1
+ (paused) starting
+ (paused) calculate first leaf
+ (paused) <--- return it
+ start next2 (paused) starting
+ (paused) (paused) calculate first leaf
+ (paused) (paused) <-- return it
+ compare leaves (paused) (paused)
+ call loop again (paused) (paused)
+ call next1 again (paused) (paused)
+ (paused) calculate next leaf (paused)
+ (paused) <-- return it (paused)
+ ... and so on ...
+
+The way we built cooperative threads here crucially relied on two heavyweight tools. First, it relied on our having a data structure (the tree zipper) capable of being a static snapshot of where we left off in the tree whose fringe we're enumerating. Second, it relied on our using mutable reference cells so that we could update what the current snapshot (that is, tree zipper) was, so that the next invocation of the `next_leaf` function could start up again where the previous invocation left off.
+
+In coming weeks, we'll learn about a different way to create threads, that relies on **continuations** rather than on those two tools. All of these tools are inter-related. As Oleg says, "Zipper can be viewed as a delimited continuation reified as a data structure." These different tools are also inter-related with monads. Many of these tools can be used to define the others. We'll explore some of the connections between them in the remaining weeks, but we encourage you to explore more.
+