From: jim Date: Sat, 25 Apr 2015 02:29:01 +0000 (-0400) Subject: cut content X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=c45dee2642b713416fb6affaebf7c9798f695ae9;ds=sidebyside cut content --- diff --git a/topics/_coroutines_and_aborts.mdwn b/topics/_coroutines_and_aborts.mdwn index 4b2b5da4..ce525b3b 100644 --- a/topics/_coroutines_and_aborts.mdwn +++ b/topics/_coroutines_and_aborts.mdwn @@ -1,189 +1,5 @@ [[!toc]] -##Same-fringe using a zipper-based coroutine## - -Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees (here I *am* drawing the labels in the diagram): - - . . - / \ / \ - . 3 1 . - / \ / \ - 1 2 2 3 - -have the same fringe: `[1; 2; 3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter new ways to approach the problem of determining when two trees have the same fringe. - - -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), move 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: - - . . - / \ / \ - . 3 1 . - / \ / \ - 1 2 2 3 - -you won't move upwards at the same steps. Keep comparing "the next leaves" until they are different, or you exhaust the leaves of only one of the trees (then again the trees have different fringes), or you exhaust the leaves of both trees at the same time, without having found leaves 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, ) - -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 = } - # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *) - val b3 : blah_record = {height = 1; weight = 3; char_tester = } - -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} -> - (* same as preceding *) - -Anyway, using record types, we might define the tree zipper interface like so: - - type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot - and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };; - - type 'a zipper = { level : 'a starred_level; 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 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) *) - - - -The following function takes an `'a tree` and returns an `'a zipper` focused on its root: - - let new_zipper (t : 'a tree) : 'a zipper = - {level = 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 botleft of t *) - let zbotleft = move_botleft (new_zipper t) - (* 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 - | 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 - | None -> (* we've finished enumerating the fringe *) - None - ) - (* 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 = - # 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 = - # 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*.