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מבוא מורחב למדעי המחשב בשפת Scheme תרגול 5 Outline • Abstraction Barriers – Fractals – Mobile • List and pairs manipulations – Insertion Sort 2 cons, car, cdr, list (cons 1 2) is a pair => (1 . 2) 2 box and pointer diagram: 1 nil = () the empty list (null in Dr. Scheme) (list 1) = (cons 1 nil) => (1) 1 3 (car (list 1 2)) 1 2 => 1 (cdr (list 1 2)) => (2) (cadr (list 1 2)) => 2 (cddr (list 1 2)) => () 4 (list 1 (list (list 2 3) 4) (cons 5 (list 6 7)) 8) 1 8 5 6 7 4 5 2 3 (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? 6 (car (car (cdr (cdr x)))) 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) cons-ing up (add-sort n (cdr s)))))) 7 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) (1 3 5 7 9) • (3 5 9 1) (1 3 5 9) • (5 9 1) (1 5 9) • (9 1) (1 9) • (1) (1) Time • () () Complexity? 8 Implementation (define (insertion-sort s) (if (null? s) null (add-sort (car s) (insertion-sort (cdr s))))) 9 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 10 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 11 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] )))) 12 Scheme triangle Constructor: (define (make-triangle a b c) (list a b c)) Selectors: (define (a-point triangle) (car triangle)) (define (b-point triangle) (cadr triangle)) (define (c-point triangle) (caddr triangle)) Predicate: (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)))) Draw: (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) 13 ((draw-line view) c a my-color)))) Points Constructor: (define (make-posn x y) (list x y)) Selectors: (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)))))) 14 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))) 15 (sierpinski (make-triangle c c-a b-c))))))))) 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 16 Mobile 17 Mobile • Left and Right branches • Constructor – (make-mobile left right) • Selectors – (left-branch mobile) – (right-branch mobile) 18 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) 19 Building mobiles (define m (make-mobile 4 8 (make-branch 4 6) 4 2 (make-branch 6 8 1 2 (make-mobile (make-branch 4 1) (make-branch 2 2))))) 20 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) 6 1 – 9 (6+1+2) 2 21 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)))) 22 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 23 Balanced mobiles • Leaf – Always Balanced • Rod 4 8 – Equal moments 5 1 – F = length x weight 6 • Mobile 1 2 – All rods are balanced = – Main rod is balanced, and both sub-mobiles • (balanced? m) 24 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)))))) 25 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 26 Mobile structures n n-1 n-2 n-3 ... T(n) = T(n-1) + (n) (for this family of mobiles) T(n) = (n2) 27 Mobile structures n/2 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 T(n) = 2T(n/2) + (n) (for this family of mobiles) T(n) = (nlogn) 28 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)) 29 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))) 30 Complexity revised • Complexity of new total-weight? • Complexity of new constructor? • Complexity of balanced? • Can we do even better? 31

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posted: | 4/26/2010 |

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