Data Abstraction: Sets Binary Search Trees

1. Data Abstraction: SetsBinary 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 3 6 9 13

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