Trees and Inductive Definitions

Transcription

1 Lecture 19 CS 1813 Discrete Mathematics Trees and Inductive Definitions 1

2 Tree What Is a Tree? a diagram or graph that branches usually from a simple stem without forming loops or polygons Merriam-Webster a type of data structure in which each element is attached to one or more elements directly beneath it Webopedia a node, together with a sequence of trees inductive definition Tree terminology subtree a node in a tree, together with its sequence of trees root the node that, with its subtrees, comprises the entire tree interior node a node with a nonempty sequence of subtrees leaf a node whose associated sequence of subtrees is empty branch a line connecting a node to its subtrees (in tree diagram) binary tree a tree with no nodes having more than 2 subtrees 2

3 What Are Trees Good For? Computing applications that use trees Databases, parsing, games, expert systems, word processors, operating systems, Search Trees a common use of trees Structures to make data quickly retrievable Each node stores a key for retrieval and a package of data associated with the key Keys are from a datatype that has an ordering Subtrees are arranged to narrow the search quickly Naïve search just look sequentially through the pile Retrieval time proportional to number of items Tree search log(n) retrieval time, n = number items 3

4 Key ordered Data anything Binary Search Tree diagram form 3663 Net Hub 5120 PDA Cam 6876 Intellimouse K Modem 4403 HotSync MB RAM 8444 Audio System 1143 InkJet 2088 LaserJet 4878 Palm Games 7268 Zip Drive 9605 Palm Pilot Each node stores key and data Left subtree contains all smaller keys Right subtree contains all larger keys Leafs just mark tree boundaries No data in a leaf To find an item start at root look left if smaller right if bigger 12 items: 4 steps, max 4

5 Binary Search Tree a formal representation data SearchTree key dat = Nub Cel key dat (SearchTree key dat) (SearchTree key dat) Key goes here Node data Left subtree (smaller keys) Type parameters key, dat Example key might be Int, for example (datatype with an ordering) dat could be any type, typically a tuple s :: SearchTree Int (String, Float, [String] ) s is a SearchTree key type Int (maybe a catalog order number) dat type tuple storing a String (product description), a Float (price), and a sequence of strings (inventory records) Nub leaf constructor Cel constructor, non-empty trees Right subtree (larger keys) 5

6 Retrieving Data from a Search Tree Found or Not Found Key sought may or may not be in search tree What to do if it s not there Deliver some sort of not-found indicator Use Maybe datatype for this data Maybe a = Just a Nothing Example, item found Just (2088, LaserJet ) Not-Found Indicator getitem :: Ord key => SearchTree key dat -> key -> Maybe (key, dat) getitem (Cel key dat smaller bigger) searchkey= if searchkey < key then (getitem smaller searchkey) else if searchkey > key then (getitem bigger searchkey) else (Just(key, dat)) getitem Nub searchkey = Nothing 6

7 Tree Induction another proof method Definitions Subtree s, t :: SearchTree key dat s t (s = t) ( k,d,left,right. (t = Cel k d left right)) ((s left) (s right)) Proper Subtree s, t :: SearchTree key dat s t ( k,d,left,right. (t = Cel k d left right)) ((s left) (s right)) Equivalent Definition: s t s t s t Tree induction P predicate parameterized over SearchTrees P(t) is a proposition whenever t :: SearchTree key dat Prove: t. ( s t. P(s)) P(t) Conclude: t. P(t) Note: {s s Nub} =, so must prove P(Nub) directly ( s Nub. P(s)) P(Nub) is equivalent to (True P(Nub)) 7

8 Properties of Trees Definition occurs in s :: SearchTree key dat, k :: key, d :: dat k occurs in s that is, k s k s ( x,d,left,right. s = (Cel x d left right)) (k = x k left k right) Definition ordered ordered s (Cel k d left right) s. (x left x < k) (x right x > k) Theorem: ordered trees have no duplicate keys P(s) ((ordered s) (x s) (s = (Cel k d left right))) (x = k (x left x right)) (x left (x k x right)) (x right (x k x left)) Proof base case: s = Nub P(s) is True because the hypothesis of P(s) is False, since it cannot be the case that Nub = (Cel k d left right) and a thing constructed by Nub cannot be the same as a thing constructed by Cel inductive case: next slide 8

9 Ordered Trees Have No Duplicate Keys inductive case Theorem: ordered trees have no duplicate keys P(s) ((ordered s) (x s) (s = (Cel k d left right))) ((x = k (x left x right)) {1} (x left (x k x right)) {2} (x right (x k x left))) {3} Note: ordered s (Cel k d left right) s. (x left x < k) (x right x > k) Proof inductive case: s = (Cel k d left right) 1. x = k (x < k) (x > k) {property of <, >, and =} x left x right {ordered, contrapositive} 2. x left x < k (x > k) {ordered, arithmetic} x k x right {arithmetic, ordered} 3. x right x > k (x < k) {ordered, arithmetic} x k x left {arithmetic, ordered} Conclusion: s. P(s) 9

10 Proving That getitem Works Theorem (getitem) ( (ordered s) k s ) getitem s k = Just (k, d) Proof P(s) ((ordered s) k s) getitem s k = Just (k, d) Base case P(Nub) ((ordered Nub) k Nub) getitem Nub k = Just(k, d) The implication is true because its hypothesis is always false k Nub would require that Nub = Cel x a left right A thing constructed by Nub cannot be the same as a thing constructed by Cel Inductive case next slide 10

11 getitem Proof inductive case getitem :: Ord key => SearchTree key dat -> key -> Maybe (key, dat) getitem (Cel key dat smaller bigger) searchkey = if searchkey < key then (getitem smaller searchkey) else if searchkey > key then (getitem bigger searchkey) else (Just(key, dat)) getitem Nub searchkey = Nothing P(s) ((ordered s) k s) d. getitem s k = Just(k, d) Inductive Case: P(s), s Nub (ordered s) k s s = Cel x a left right s Nub Case 1. x = k getitem s k = Just(k, a) x = k (k < x) (k > x) Case 2. k < x k left s is ordered d. getitem left k = Just(k, d) induction hyp: left s P(left) getitem s k = getitem left k searchkey < key branch of if-then getitem s k = Just(k, d) both sides = (getitem left k) Case 3. k > x like Case 2, but use P(right) 11

12 End of Lecture 12