Data Abstraction: Sets

1
Data Abstraction Sets

• CMSC 11500
• Introduction to Computer Programming
• October 21, 2002

2
Administration

• Midterm Wednesday in-class
• Hand evaluations
• Structures
• Self-referential structures lists, structures of
  structs
• Data definitions, Templates, Functions
• Recursion in Fns follow recursion in data def
• Recursive/Iterative processes
• Based on lecture/notes/hwk
• Not book details
• Extra office hrs Tues 3-5, RY 178

3
Roadmap

• Recap Structures of structures
• Data abstraction
• Black boxes redux
• Objects as functions on them
• Example Sets
• Operations Member-of?, Adjoin, Intersection
• Implementations and Efficiency
• Unordered Lists
• Ordered Lists
• Binary Search Trees
• Summary

4
Recap Structures of Structures

• Family trees
• (Multiply) self-referential structures
• Data Definition -gt Template -gt Function
• (define-struct ft (name eye-color mother father))
• Where name, eye-color symbol mother, father
  family tree
• A family-tree 1) unknown,
• 2) (make-ft name eye-color
  mother father)

5
Template -gt Function
(define (bea aft) (cond ((eq? unknown aft)
f) ((ft? aft) (let ((bea-m (bea
(ft-mother aft))) (bea-f (bea (ft-father
aft)))) (cond ((eq ?(ft-eye-color aft)
blue) (ft-name aft)) ((symbol? bea-m)
bea-m) ((symbol? bea-f) bea-f) (else f))))))
(define (fn-for-ft aft) (cond ((eq? unknown
aft)..) ((ft? aft) (cond
(ft-eye-color aft) (ft-name aft).
(fn-for-ft (ft-mother aft))..
(fn-for-ft (ft-father aft))
6
Data Abstraction

• Analogous to procedural abstraction
• Procedure is black box
• Dont care about implementation as long as
  behaves as expected contract, purpose
• Data object as black box
• Dont care about implementation as long as
  behaves as expected

7
Abstracting Data

• Compound data objects
• Points, families, rational numbers, sets
• Data has many facets
• Also many possible representations
• Key Data object defined by operations
• E.g. points distance, slope, etc
• Any implementation acceptable if performs
  operations specified

8
Data Abstraction Sets

• Set Collection of distinct objects
• Venn Diagrams
• Defining functions
• Element-of? true if element is member of set
• Adjoin Adds element to set
• Union Set containing elements of input sets
• Intersection Set of elements in both input sets
• Any implementation of these functions defines a
  set data object

9
Sets Representations Efficiency

• Many possible representations
• Unordered lists
• Ordered lists
• Binary Search Trees
• How choose among representations?
• Efficiency Order of growth
• Tradeoffs in operations

10
Sets as Unordered Lists

• A Set-of-numbers is
• 1) ()
• 2) (cons n set-of-numbers)
• Where n is number
• Set Each element appears once
• Base template
• (define (fn-for-set set)
• (cond ((null? set) )
• (else ((car set) .(fn-for-set (cdr
  set)))))

11
Member-of?
(define (member-of x set) member-of number set
-gt boolean true if x in set, false
otherwise (cond ((null? set) f) ((eq? (car
set) x) t) (else (member-of x (cdr
set))))) Order of growth Length of list O(n)
12
Member-of? Hand-Evaluation
(define (member-of x set) member-of number set
-gt boolean true if x in set, false
otherwise (cond ((null? set) f) ((eq? (car
set) x) t) (else (member-of x (cdr set)))))
(member-of 3 (1 2 3 4))
(cond ((null? (1 2 3 4)) f) ((eq?
(car (1 2 3 4)) 3) t) (else (member-of 3 (cdr
(1 2 3 4)))))
(cond (f f) ((eq? (car (1 2 3 4)) 3)
t) (else (member-of 3 (cdr (1 2 3 4)))))
13
Hand-Evaluation Contd
(cond (f f) ((eq? 1 3) t) (else
(member-of 3 (cdr (1 2 3 4)))))
(cond (f f) (f t) (else (member-of
3 (cdr (1 2 3 4)))))
(cond (f f) (f t) (else (member-of
3 (2 3 4))))
(cond ((null? (2 3 4)) f) ((eq? (car
(2 3 4)) 3) t) (else (member-of 3 (cdr (2 3
4)))))
14
Hand-Evaluation Contd
(cond ((f f) ((eq? (car (3 4) 3)
t) (else (member-of 3 (3 4))))
(cond (f f) ((eq? (car (2 3 4) 3)
t) (else (member-of 3 (cdr (2 3 4)))))
(cond ((f f) ((eq? (car (3 4) 3)
t) (else (member-of 3 (3 4))))
(cond (f f) ((eq? 2 3) t) (else
(member-of 3 (cdr (2 3 4)))))
(cond ((f f) ((eq? 3 3) t) (else
(member-of 3 (3 4))))
(cond (f f) (f t) (else (member-of
3 (cdr (2 3 4))))
(cond ((f f) (t t) (else
(member-of 3 (3 4))))
(cond ((f f) (f t) (else
(member-of 3 (3 4))))
(cond ((null? (3 4) f) ((eq? (car (3
4) 3) t) (else (member-of 3 (cdr (3 4)))))
t
15
Hand-Evaluation Recursion Only
(member-of 3 (2 3 4))
(member-of 3 (3 4))
((eq? 3 3) t) t
16
Adjoin

• Question Single occurrence of element
• Maintain at adjoin? Always insert?
• Add condition to maintain invariant (one
  occurrence)
• Order of Growth Member-of? test O(n)

(define (adjoin x set) (cond ((null? set)
(cons x set)) ((eq? (car set) x) set)
(else (cons (car set) (adjoin x
(cdr set)))))
(define (adjoin x set) (if (member-of? x
set) set (cons x set)))
17
Intersection
(define (intersection set1 set2) intersection
set set -gt set (cond ((null? set1) ())
((null? set2) ()) ((member-of? (car set1)
set2) (cons (car set1)
(intersection (cdr set1) set2)) (else
(intersection (cdr set1) set2)))

• Order of growth
• Test every member of set1 in set2
• Member O(n) Length of set1 O(n) ? O(n2)

18
Alternative Ordered Lists

• Set-of-numbers
• 1) ()
• 2) (cons n set-of-numbers)
• Where n is a number, and n lt all numbers in son
• Maintain constraint
• Anywhere add element to set

19
Adjoin
(define (adjoin x set) (cond ((null? set) (cons x
()) ((eq? (car set) x) set) ((lt 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
20
Sets as Binary Search Trees

• Bst 1) ()
• 2) (make-bstn val left right)
• Where val is number left, right are bst
• All vals in left branch less than current, right
  gtr
• (define-struct bstn (val left right))

7
5
11
3
6
9
13
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Adjoin BST
(define (adjoin x set) (cond ((null? set)
(make-bstn x () ()) (( x (bstn-val set))
set) ((lt x (bstn-val set)) (make-bstn
(bstn-val set) (adjoin x (bstn-left set))
(bstn-right set))) ((gt x (bstn-val set))
(make-bstn (bstn-val set) (bstn-left
set) (adjoin x (bstn-right set))))
22
BST Sets

• Analysis
• If balanced tree
• Each branch reduces tree size by half
• Successive halving -gt O(log n) growth

23
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

24
Next Time

• Midterm
• Hand evaluations
• Structures
• Self-referential structures lists, structures of
  structs
• Data definitions, Templates, Functions
• Recursion in Fns follow recursion in data def