From: Jim Pryor Date: Sun, 3 Oct 2010 07:57:47 +0000 (-0400) Subject: add trees to advanced X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=58137a1e8e52e3c15790ab3209b4f217dcdd1ba4 add trees to advanced Signed-off-by: Jim Pryor --- diff --git a/miscellaneous_lambda_challenges_and_advanced_topics.mdwn b/miscellaneous_lambda_challenges_and_advanced_topics.mdwn index f05f9fe3..a4ba9854 100644 --- a/miscellaneous_lambda_challenges_and_advanced_topics.mdwn +++ b/miscellaneous_lambda_challenges_and_advanced_topics.mdwn @@ -491,4 +491,198 @@ can use. 5. **Implementing (self-balancing) trees** - more... + In [[Assignment3]] we proposed a very ad-hoc-ish implementation of trees. + + Think about how you'd implement them in a more principled way. You could + use any of the version 1 -- version 5 implementation of lists as a model. + + To keep things simple, I recommend starting with the version 3 pattern. And + stick to binary trees. + + There are two kinds of trees to think about. In one sort of tree, it's only + the tree's *leaves* that are labeled: + + + + / \ + + 3 + / \ + 1 2 + + Linguists often use trees of this sort. The inner, non-leaf nodes of the +tree have associated values. But what values they are can be determined from +the structure of the tree and the values of the node's left and right children. +So the inner node doesn't need its own independent label. + + In another sort of tree, the tree's inner nodes are also labeled: + + 4 + / \ + 2 5 + / \ + 1 3 + + When you want to efficiently arrange an ordered collection, so that it's + easy to do a binary search through it, this is the way you usually structure + your data. + + These latter sorts of trees can helpfully be thought of as ones where + *only* the inner nodes are labeled. Leaves can be thought of as special, + dead-end branches with no label: + + .4. + / \ + 2 5 + / \ / \ + 1 3 x x + / \ / \ + x x x x + + In our earlier discussion of lists, we said they could be thought of as + data structures of the form: + + Empty_list | Non_empty_list (its_head, its_tail) + + And that could in turn be implemented in v2 form as: + + the_list (\head tail. non_empty_handler) empty_handler + + Similarly, the leaf-labeled tree: + + + + / \ + + 3 + / \ + 1 2 + + can be thought of as a data structure of the form: + + Node (its_left_subtree, its_right_subtree) | Leaf (its_label) + + and that could be implemented in v2 form as: + + the_tree (\left right. node_handler) (\label. lead_handler) + + And the node-labeled tree: + + .4. + / \ + 2 5 + / \ / \ + 1 3 x x + / \ / \ + x x x x + + can be thought of as a data structure of the form: + + Node (its_left_subtree, its_label, its_right_subtree) | Leaf + + and that could be implemented in v2 form as: + + the_tree (\left label right. node_handler) leaf_result + + + What would correspond to "folding" a function f and base value z over a + tree? Well, if it's an empty tree: + + x + + we should presumably get back z. And if it's a simple, non-empty tree: + + 1 + / \ + x x + + we should expect something like f z 1 z, or f label_of_this_node . (It's not important what order we say f has to take its arguments + in.) + + A v3-style implementation of node-labeled trees, then, might be: + + let empty_tree = \f z. z in + let make_tree = \left label right. \f z. f (left f z) label (right f z) in + ... + + Think about how you might implement other tree operations, such as getting +the label of the root (topmost node) of a tree; extracting the left subtree of +a node; and so on. + + Think about different ways you might implement v3-style trees. + + If you had one tree and wanted to make a larger tree out of it, adding in a +new element, how would you do that? + + When using trees to represent linguistic structures, one doesn't have +latitude about *how* to build a larger tree. The linguistic structure you're +trying to represent will determine where the new element should be placed, and +where the previous tree should be placed. + + However, when using trees as a computational tool, one usually does have +latitude about how to structure a larger tree---in the same way that we had the +freedom to implement our sets with lists whose members were just appended in +the order we built the set up, or instead with lists whose members were ordered +numerically. + + When building a new tree, one strategy for where to put the new element and +where to put the existing tree would be to always lean towards a certain side. +For instance, to add the element 2 to the tree: + + 1 + / \ + x x + + we might construct the following tree: + + 1 + / \ + x 2 + / \ + x x + + or perhaps we'd do it like this instead: + + 2 + / \ + x 1 + / \ + x x + + However, if we always leaned to the right side in this way, then the tree +would get deeper and deeper on that side, but never on the left: + + 1 + / \ + x 2 + / \ + x 3 + / \ + x 4 + / \ + x 5 + / \ + x x + + and that wouldn't be so useful if you were using the tree as an arrangement +to enable *binary searches* over the elements it holds. For that, you'd prefer +the tree to be relatively "balanced", like this: + + .4. + / \ + 2 5 + / \ / \ + 1 3 x x + / \ / \ + x x x x + + Do you have any ideas about how you might efficiently keep the new trees +you're building pretty "balanced" in this way? + + This is a large topic in computer science. There's no need for you to learn +any of the various strategies that they've developed for doing this. But +thinking in broad brush-strokes about what strategies might be promising will +help strengthen your understanding of trees, and useful ways to implement them +in a purely functional setting like the lambda calculus. + + + +