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(load "displaylib.scm")
(title "Exercise 2.65")
(print "
Use the results of exercises 2.63 and 2.64 to give (n) implementations of union-set and intersection-set for sets implemented as (balanced) binary trees.
; The implementation of union-set and intersection set in ϴ(n) can be done using
; an intermediate ordered list representation.
; We have a ϴ(n) implementation for union and intersection for ordered list
; We have a ϴ(n) for tree -> ordered list
; We have a ϴ(n) for list -> balanced tree
; -- transform a tree into an ordered list
(define (tree->list tree)
(define (copy-to-list tree result-list)
(if (null? tree)
(copy-to-list (left-branch tree)
(cons (entry tree)
(copy-to-list (right-branch tree)
(copy-to-list tree '()))
; Transform an ordered list into a balanced tree
(define (list->tree elements)
(car (partial-tree elements (length elements))))
(define (partial-tree elts n)
(if (= n 0)
(cons '() elts)
(let ((left-size (quotient (- n 1) 2)))
(let ((left-result (partial-tree elts left-size)))
(let ((left-tree (car left-result))
(non-left-elts (cdr left-result))
(right-size (- n (+ left-size 1))))
(let ((this-entry (car non-left-elts))
(right-result (partial-tree (cdr non-left-elts)
(let ((right-tree (car right-result))
(remaining-elts (cdr right-result)))
(cons (make-tree this-entry left-tree right-tree)
; ϴ(n) union for two ordered list
(define (ordlist-union-set e f)
(cond ((null? e) f)
((null? f) e)
(let ((x1 (car e)) (x2 (car f)))
(cond ((= x1 x2) (cons x1 (ordlist-union-set (cdr e) (cdr f))))
((< x1 x2) (cons x1 (ordlist-union-set (cdr e) f)))
((> x1 x2) (cons x2 (ordlist-union-set e (cdr f)))))))))
; ϴ(n) intersection for two ordered list
(define (ordlist-intersection-set set1 set2)
(if (or (null? set1) (null? set2))
(let ((x1 (car set1)) (x2 (car set2)))
(cond ((= x1 x2)
(cons x1
(ordlist-intersection-set (cdr set1)
(cdr set2))))
((< x1 x2)
(ordlist-intersection-set (cdr set1) set2))
((< x2 x1)
(ordlist-intersection-set set1 (cdr set2)))))))
(define (union-set set1 set2)
(let ((ordlist-set1 (tree->list set1))
(ordlist-set2 (tree->list set2)))
(list->tree (ordlist-union-set ordlist-set1 ordlist-set2))))
(define (intersection-set set1 set2)
(let ((ordlist-set1 (tree->list set1))
(ordlist-set2 (tree->list set2)))
(list->tree (ordlist-intersection-set ordlist-set1 ordlist-set2))))