Recitation 5
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Transcript Recitation 5
מבוא מורחב למדעי המחשב
בשפת Scheme
תרגול 5
Outline
• Let*
• List and pairs manipulations
– Insertion Sort
• Abstraction Barriers
– Fractals
– Mobile
2
let*
(let* ((<var1> <exp1>)…
(<varn> <expn>))
<body>)
is (almost) equivalent to
(let ((<var1> <exp1>))
(let* ((<var2> <exp2>)…
(<varn> <expn>))
<body>))
3
let vs. let*
(let ((x 2) (y 3))
(let ((x 7)
(z (+ x y)))
(* z x)))
==> 35
4
let vs. let*
(let ((x 2) (y 3))
(let* ((x 7)
(z (+ x y)))
(* z x)))
==> 70
5
cons, car, cdr, list
(cons 1 2) is a pair => (1 . 2)
box and pointer diagram:
2
1
nil = () the empty list (null in Dr. Scheme)
(list 1) = (cons 1 nil) => (1)
1
6
(car (list 1 2))
=> 1
(cdr (list 1 2))
=> (2)
(cadr (list 1 2))
=> 2
(cddr (list 1 2))
=> ()
1
2
7
(list 1 (list (list 2 3) 4) (cons 5 (list 6 7)) 8)
8
1
5
6
7
4
2
3
8
(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))))
5
4
1
3
2
How to reach the 3 with cars and cdrs?
(car (car (cdr (cdr x))))
9
cdr-ing down a list
cons-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)
cdr-ing down
(define (add-sort n s)
(cond ((null? s) (list n))
((< n (car s)) (cons n s))
(else (cons (car s)
(add-sort n (cdr s))))))
cons-ing up
10
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?
11
Implementation
(define (insertion-sort s)
(if (null? s) null
(add-sort (car s)
(insertion-sort (cdr s)))))
12
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
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sub-division.
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
14
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]
)
)
))))
15
Scheme triangle
Constructor:
Selectors:
Predicate:
Draw:
(define
(define
(define
(define
(make-triangle a b
(a-point triangle)
(b-point triangle)
(c-point triangle)
c) (list a b c))
(car triangle))
(cadr 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))))
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Points
Constructor:
Selectors:
(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))))))
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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 (make-triangle a a-b c-a)))
(sierpinski (make-triangle b a-b b-c)))
(sierpinski (make-triangle c c-a b-c)))))))))
18
Abstraction barriers
Programs that use Triangles
Triangles in problem domain
too-small? draw-triangle
Triangles as lists of three points
make-triangle a-point b-point c-point
Points as lists of two coordinates (x,y)
make-posn posn-x posn-y
Points as lists
cons list car cdr
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Mobile
20
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)
22
Building mobiles
(define m
(make-mobile
4
8
(make-branch 4 6)
4
(make-branch
2
6
8
1
2
(make-mobile
(make-branch 4 1)
(make-branch 2 2)))))
23
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 (branch-structure
(left-branch mobile)))
(total-weight (branch-structure
(right-branch mobile)))
)))
(define (atom? x)
(and (not (pair? x)) (not (null? x))))
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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
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Balanced mobiles
• Leaf
– Always Balanced
• Rod
4
8
– Equal moments
– F = length x weight
• Mobile
5
1
6
1
2
– All rods are balanced =
– Main rod is balanced, and both sub-mobiles
• (balanced? m)
27
balanced?
(define (balanced? mobile)
(or (atom? mobile)
(let ((l (left-branch mobile))
(r (right-branch mobile)))
(and
(= (* (branch-length l)
(total-weight
(branch-structure l)))
(* (branch-length r)
(total-weight
(branch-structure r))))
(balanced? (branch-structure l))
(balanced? (branch-structure r))))))
28
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)
T(n) = (n2)
(for this family of mobiles)
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Mobile structures
n/2
n/4
n/8 n/8
n/4
n/8 n/8
T(n) = 2T(n/2) + (n)
T(n) = (nlogn)
n/2
n/4
n/8 n/8
n/4
n/8 n/8
(for this family of mobiles)
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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)))
33
Complexity revised
•
•
•
•
Complexity of new total-weight?
Complexity of new constructor?
Complexity of balanced?
Can we do even better?
34