32
בבבב בבבבב בבבבב בבבבב בבבבScheme ללללל4
מבוא מורחב למדעי המחשב Schemeבשפת
4תרגול
Outline
• Let*
• List and pairs manipulations– Insertion Sort
• Abstraction Barriers – Fractals– Mobile
2
let*
(let* ((<var1> <exp1>)…(<varn> <expn>))
<body>)
is equivalent to
(let ((<var1> <exp1>)) (let* ((<var2> <exp2>)… (<varn> <expn>)) <body>))
3
4
cons, car, cdr, list
(cons 1 2) is a pair => (1 . 2)box and pointer diagram:
nil = () the empty list (null in Dr. Scheme)
(list 1) = (cons 1 nil) => (1)
1
2
1
5
(car (list 1 2)) => 1(cdr (list 1 2)) => (2)(cadr (list 1 2)) => 2(cddr (list 1 2)) => ()
1 2
6
(list 1 (list (list 2 3) 4) (cons 5 (list 6 7)) 8)
1
4
32
5 6 7
8
7
(5 4 (3 2) 1)(list 5 4 (list 3 2) 1) (cons 5 (cons 4 (cons (cons 3 (cons 2 nil)) (cons 1 null))))
15 4
23
How to reach the 3 with cars and cdrs?
)car (car (cdr (cdr x)))(
8
cdr-ing down a listcons-ing up a list
(add-sort 4 (list 1 3 5 7 9))(1 3 4 5 7 9)(add-sort 5 ‘())(5)(add-sort 6 (list 1 2 3))(1 2 3 6)
(define (add-sort n s) (cond ((null? s) ) ((< n (car s)) ) (else )))
(list n) (cons n s) (cons (car s)
(add-sort n (cdr s)))cons-ing up
cdr-ing down
9
Insertion sort
• An empty list is already sorted• To sort a list with n elements:
– Drop the first element– Sort remaining n-1 elements (recursively)– Insert the first element to correct place
• (7 3 5 9 1)• (3 5 9 1)• (5 9 1)• (9 1)• (1)• ()
(1 3 5 7 9)(1 3 5 9)(1 5 9)(1 9)(1)()
Time Complexity?
10
Implementation
(define (insertion-sort s)
(if (null? s) null
(add-sort (car s)
(insertion-sort (cdr s)))))
11
Fractals
Definitions: • A mathematically generated pattern that is reproducible at any magnification or reduction. • A self-similar structure whose geometrical and topographical features are recapitulated in miniature on finer and finer scales.• An algorithm, or shape, characterized by self-similarity and produced by recursive sub-division.
12
Sierpinski triangle
• Given the three endpoints of a triangle, draw the triangle• Compute the midpoint of each side• Connect these midpoints to each other, dividing the given triangle into four triangles• Repeat the process for the three outer triangles
13
Sierpinski triangle – Scheme version
(define (sierpinski triangle)
(cond
((too-small? triangle) #t)
(else
(draw-triangle triangle)
(sierpinski [outer triangle 1] )
(sierpinski [outer triangle 2] )
(sierpinski [outer triangle 3] ))))
14
Scheme triangle(define (make-triangle a b c) (list a b c))(define (a-point triangle) (car triangle)) (define (b-point triangle) (cadr triangle)) (define (c-point triangle) (caddr triangle))
(define (too-small? triangle) (let ((a (a-point triangle)) (b (b-point triangle)) (c (c-point triangle))) (or (< (distance a b) 2)
(< (distance b c) 2) (< (distance c a) 2))))
(define (draw-triangle triangle) (let ((a (a-point triangle)) (b (b-point triangle)) (c (c-point triangle))) (and ((draw-line view) a b my-color) ((draw-line view) b c my-color) ((draw-line view) c a my-color))))
Constructor:
Selectors:
Predicate:
Draw:
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Points(define (make-posn x y) (list x y))(define (posn-x posn) (car posn)) (define (posn-y posn) (cadr posn))
(define (mid-point a b) (make-posn (mid (posn-x a) (posn-x b)) (mid (posn-y a) (posn-y b))))
(define (mid x y) (/ (+ x y) 2))
(define (distance a b) (sqrt (+ (square (- (posn-x a) (posn-x b))) (square (- (posn-y a) (posn-y b))))))
Constructor:
Selectors:
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Sierpinski triangle – Scheme final version
(define (sierpinski triangle)
(cond
((too-small? triangle) #t)
(else
(let ((a (a-point triangle))
(b (b-point triangle))
(c (c-point triangle)))
(let ((a-b (mid-point a b))
(b-c (mid-point b c))
(c-a (mid-point c a)))
(and
(draw-triangle triangle)
(sierpinski )
(sierpinski )
(sierpinski )))))))
(make-triangle a a-b c-a))
(make-triangle b a-b b-c))
(make-triangle c c-a b-c))
17
Abstraction barriers
Programs that use Triangles
too-small? draw-triangle
make-posn posn-x posn-y
cons list car cdr
Triangles in problem domain
Points as lists of two coordinates (x,y)
Points as lists
make-triangle a-point b-point c-point
Triangles as lists of three points
18
Mobile
19
Mobile
• Left and Right branches
• Constructor– (make-mobile left right)
• Selectors– (left-branch mobile)– (right-branch mobile)
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Branch
• Length and Structure– Length is a number– Structure is…
• Another mobile• A leaf (degenerate mobile)
– Weight is a number
• Constructor– (make-branch length structure)
• Selectors– (branch-length branch)– (branch-structure branch)
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Building mobiles
61 2
(define m
(make-mobile
(make-branch 4 6)
(make-branch
8
(make-mobile
(make-branch 4 1)
(make-branch 2 2)))))
24
4 8
22
Mobile weight
• A leaf’s weight is its value
• A mobile’s weight is: – Sum of all leaves =– Sum of weights on both sides
• (total-weight m)
– 9 (6+1+2)6
1 2
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Mobile weight
(define (total-weight mobile)
(if (atom? mobile) mobile
(+ (total-weight
)
(total-weight
)
)))
(define (atom? x)
(and (not (pair? x)) (not (null? x))))
(branch-structure
(left-branch mobile))(branch-structure
(right-branch mobile))
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Complexity Analysis
• What does “n” represent?– Number of weights?– Number of weights, sub-mobiles and branches?– Number of pairs?– All of the above?
• Analysis (n)– Depends on mobile’s size, not structure
25
Balanced mobiles
• Leaf– Always Balanced
• Rod– Equal moments– F = length x weight
• Mobile– All rods are balanced =– Main rod is balanced, and both sub-mobiles
• (balanced? m)
6
12
15
4 8
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balanced?(define (balanced? mobile) (or (atom? mobile) (let ((l (left-branch mobile)) (r (right-branch mobile))) (and (= ) (balanced? ) (balanced? )))))
(* (branch-length l) (total-weight (branch-structure l))) (* (branch-length r) (total-weight (branch-structure r)))
(branch-structure l) (branch-structure r)
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Complexity
• Worst case scenario for size n– Need to test all rods– May depend on mobile structure
• Upper bound– Apply total-weight on each sub-mobile– O(n2)
• Lower bound
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Mobile structures
n
n-1
n-2
n-3
. . .
T(n) = T(n-1) + (n) (for this family of mobiles)
T(n) = (n2)
29
Mobile structures
n/2
T(n) = 2T(n/2) + (n) (for this family of mobiles)
T(n) = (nlogn)
n/2
n/4 n/4 n/4 n/4
n/8 n/8 n/8 n/8 n/8 n/8 n/8 n/8
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Implementation
Constructors(define (make-mobile left right) (list left right))(define (make-branch length structure) (list length structure))
Selectors(define (left-branch mobile) (car mobile))(define (right-branch mobile) (cadr mobile))(define (branch-length branch) (car branch))(define (branch-structure branch) (cadr branch))
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Preprocessing the data
• Calculate weight on creation:– (define (make-mobile left right) (list left right (+ (total-weight (branch-structure left)) (total-weight (branch-structure right)))))
• New Selector:– (define (mobile-weight mobile) (caddr mobile))
• Simpler total-weight:– (define (total-weight mobile) (if (atom? mobile) mobile (mobile-weight mobile)))
32
Complexity revised
• Complexity of new total-weight?
• Complexity of new constructor?
• Complexity of balanced?
• Can we do even better?
