Data Abstraction: Sets Binary Search Trees

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Data Abstraction SetsBinary Search Trees

• CMSC 11500
• Introduction to Computer Programming
• October 30, 2002

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Roadmap

• Recap Binary Trees
• Data abstractionSets
• Objects as functionsMember-of?, Adjoin,
  Intersection
• Implementations Binary Search Trees
• Data Definition
• Invariants
• Template
• Functions
• Analysis Efficiency
• Summary

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Recap Binary Trees

• Binary trees
• (Multiply) self-referential structures
• Data Definition -gt Template -gt 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))

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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

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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-vals of all nodes in
  (bt-left n) are less than (bt-val n) and all node
  values in (bt-right n) are greater.

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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 gt anything to
  its left, lt than anything to its right

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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))
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Sets as Binary Search Trees

• 3,5,6,7,9,11,13
• Balanced same nodes in left right
• Unbalanced anything else

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5
11
3
6
9
13
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Element-of?

• Contract
• element-of? number bst -gt boolean
• Purpose
• To determine if element is in set

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Element-of?
(define (element-of? num abst) (cond ((eq?
abst f) f) ((bt? abst) (cond (( num
(bt-val abst)) t) ((lt num (bt-val
abst)) (element-of? num (bt-left abst)))
((gt num (bt-val abst)) (element-of? num
(bt-right abst)))))
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Adjoin

• Contract
• adjoin number bst -gt bst
• Purpose
• To create a new set with members of original
  set and new element

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Adjoin BST
(define (adjoin x abst) (cond ((null? abst)
(make-bt x f f) (( x (bt-val abst))
abst) ((lt x (bt-val abst)) (make-bt
(bt-val abst) (adjoin x (bt-left abst))
(bt-right abst))) ((gt x (bt-val abst))
(make-bt (bt-val abst) (bt-left abst)
(adjoin x (bt-right abst))))
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Intersection BST

• Contract
• intersection bst bst -gt bst
• Purpose
• To build a set including items found in both
  sets

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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)
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BST Sets

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

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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

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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

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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