מבוא מורחב למדעי המחשב בשפת Scheme תרגול 5

Outline Abstraction Barriers –Fractals –Mobile List and pairs manipulations –Insertion Sort 2

3 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

4 (car (list 1 2)) => 1 (cdr (list 1 2)) => (2) (cadr (list 1 2)) => 2 (cddr (list 1 2)) => () 1 2

5 (list 1 (list (list 2 3) 4) (cons 5 (list 6 7)) 8)

6 (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)))) How to reach the 3 with cars and cdrs? (car (car (cdr (cdr x))))

7 cdr-ing down a list cons-ing up a list (add-sort 4 (list ))  ( ) (add-sort 5 ‘())  (5) (add-sort 6 (list 1 2 3))  ( ) (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

8 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 ( ) ( ) (5 9 1) (9 1) (1) () ( ) ( ) (1 5 9) (1 9) (1)(1) () Time Complexity?

9 Implementation (define (insertion-sort s) (if (null? s) null (add-sort (car s) (insertion-sort (cdr s)))))

10 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.

11 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

12 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] ))))

13 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:

14 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:

15 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 ))))))) (make-triangle a a-b c-a)) (make-triangle b a-b b-c)) (make-triangle c c-a b-c))

16 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

17 Mobile

18 Mobile Left and Right branches Constructor –(make-mobile left right) Selectors –(left-branch mobile) –(right-branch mobile)

19 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)

20 Building mobiles (define m (make-mobile (make-branch 4 6) (make-branch 8 (make-mobile (make-branch 4 1) (make-branch 2 2))))) 24 48

21 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

22 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))

23 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

24 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)

25 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)

26 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(n 2 ) Lower bound

27 Mobile structures n n-1 n-2 n-3... T(n) = T(n-1) +  (n)(for this family of mobiles) T(n) =  (n 2 )

28 Mobile structures n/2 T(n) = 2T(n/2) +  (n)(for this family of mobiles) T(n) =  (nlogn) n/2 n/4 n/8

29 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))

30 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)))

31 Complexity revised Complexity of new total-weight? Complexity of new constructor? Complexity of balanced? Can we do even better?