1. Data Abstraction: Sets Binary Search Trees CMSC 11500 Introduction to Computer Programming October 30, 2002

2. Roadmap Recap: Binary Trees Data abstraction:Sets Objects as functions:Member-of?, Adjoin, Intersection Implementations: Binary Search Trees »Data Definition »Invariants »Template »Functions »Analysis & Efficiency Summary

3. Recap: Binary Trees Binary trees: –(Multiply) self-referential structures –Data Definition -> Template -> Function (define-struct bt (val left right)) –Where val: number; left, right: binary-tree A binary-tree: 1) ‘unknown, – 2) (make-bt val left right) (define (fn-for-bt abt) –(cond ((eq? abt #f)…) – ((bt? abt) – (cond (…(bt-val abt)…) ….(fn-for-bt (bt-left abt))… ….(fn-for-bt (bt-right abt))…

4. Data Abstraction: Sets Set: collection of objects Defined by operations that can be performed –Set operations: Element-of?, Adjoin, Union, Intersection, Set difference Many possible concrete implementations –Unordered lists –Ordered lists –Binary trees –Binary search trees

5. Data Definition Binary search tree (bst) is –#f or, –(make-bt val left right) Where val: number; left, right: bst (define-struct bt (val left right)) Invariant: –For a node n, the bt-val’s of all nodes in (bt- left n) are less than (bt-val n) and all node values in (bt-right n) are greater.

6. Data Invariants “invariant”: if it is true of the input, must be true of the output Invariant must be maintained by all functions that operate on the type –E.g. adjoin: new element must be >= anything to its left, < than anything to its right

7. Template (define (fn-for-bst abst) (cond ((eq? abst #f)..) ((bt? abst) (cond (( …bt-val abst)…) …(fn-for-bst (bt-left abst)).. …(fn-for-bst (ft-right abst))…

8. Sets as Binary Search Trees {3,5,6,7,9,11,13} Balanced: same # nodes in left & right Unbalanced: anything else 7 5 11 36913

9. Element-of? Contract: –;; element-of?: number bst -> boolean Purpose: –;; To determine if element is in set

10. Element-of? (define (element-of? num abst) (cond ((eq? abst #f) #f) ((bt? abst) (cond ((= num (bt-val abst)) #t) ((< num (bt-val abst)) (element-of? num (bt-left abst))) ((> num (bt-val abst)) (element-of? num (bt-right abst)))))

11. Adjoin Contract: –;; adjoin: number bst -> bst Purpose –;; To create a new set with members of original set and new element

12. Adjoin: BST (define (adjoin x abst) (cond ((null? abst) (make-bt x #f #f) ((= x (bt-val abst)) abst) ((< x (bt-val abst)) (make-bt (bt-val abst) (adjoin x (bt-left abst)) (bt-right abst))) ((> x (bt-val abst)) (make-bt (bt-val abst) (bt-left abst) (adjoin x (bt-right abst))))

13. Intersection: BST Contract: –;;intersection: bst bst -> bst Purpose: –;;To build a set including items found in both sets

14. Intersection (define (intersection bst1 bst2) (intersect-iter bst1 bst2 #f) (define (intersect-iter bst1 bst2 bst-new) (cond ((or (not bst1)(not bst2)) bst-new) ((element-of? (bt-val bst2) bst1) (intersect-iter (bt-right bst2) bst1 (intersect-iter (bt-left bst2) bst1 (adjoin (bt-val bst2) bst-new) (else (intersect-iter (bt-right bst2) bst1 (intersect-iter (bt-left bst2) bst1 bst-new)

15. BST Sets Analysis: –If balanced tree Each branch reduces tree size by half Successive halving -> O(log n) growth Element-of?: O(log n) Adjoin: O(log n)

16. Summary Data objects –Defined by operations done on them –Abstraction: Many possible implementations of operations All adhere to same contract, purpose Different implications for efficiency

17. Alternative: Ordered Lists Set-of-numbers: –1) ‘() –2) (cons n set-of-numbers) Where n is a number, and n <= all numbers in son Maintain constraint: –Anywhere add element to set

18. Adjoin (define (adjoin x set) (cond ((null? set) (cons x ‘()) ((eq? (car set) x) set) ((< x (car set)) (cons x set)) (else (cons (car set) (adjoin x (cdr set)))))) Note: New invariant adds condition Order of Growth: On average, check half