/ | \
leaf 1 leaf 2 leaf 3
-We'll call this new untargetted tree `node 20`. The result of moving upward from our previous targetter 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 `*`:
+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 = ...;
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 built a list containing the rest of each tree's 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.