1 מבוא מורחב למדעי המחשב בשפת Scheme תרגול 11

2 Outline Mutable list structure RPN calculator Vectors and sorting

3 (define (set-to-wow! x) (set-car! (car x) 'wow) x) (define x (list 'a 'b)) (define z1 (cons x x)) (set-to-wow! z1) ((wow b) wow b) eq? point to the same object, equal? same content (eq? (car z1) (cdr z1)) is true (eq? (car z2) (cdr z2)) is false (define z2 (cons (list 'a 'b) (list 'a 'b))) (set-to-wow! z2) ((wow b) a b)

4 Map without list copying (define (map! f s) (if (null? s) 'done (begin (set-car! s (f (car s))) (map! f (cdr s))))) (define s '( )) (map! square s) => done s => ( )

5 append! (define (append! x y) (set-cdr! (last-pair x) y) x) where: (define (last-pair x) (if (null? (cdr x)) x (last-pair (cdr x))))

6 append vs. append! (define x ‘(a b)) (define y ‘(c d)) (define z (append x y)) z ==> (a b c d) (cdr x) ==> ? (define w (append! x y)) w ==> (a b c d) (cdr x) ==> ?

7 (define (last-pair x) (if (null? (cdr x)) x (last-pair (cdr x)))) (define (make-cycle x) (set-cdr! (last-pair x) x) x) (define z (make-cycle (list 'a 'b 'c))) Cycle

8 What happens? (last-pair z) => a b c z

9 Examine the list and determine whether it contains a cycle, whether a program that tried to find the end of the list by taking successive cdr s would go into an infinite loop (define (is-cycle? c) (define (loop fast slow) (cond ((null? fast) #f) ((null? (cdr fast)) #f) ((eq? (cdr fast) slow) #t) (else (loop (cddr fast) (cdr slow))))) (loop c c)) is-cycle?

10 Bounded Counter A bounded counter can be incremented or decremented in steps of 1, until upper and lower bounds are reached. syntax: (make-bounded-counter init bottom top) example: (define c (make-bounded-counter 3 1 5)) (counter-inc! c)  4 (counter-inc! c)  5 (counter-dec! c)  4

11 List Implementation Constructor: (define (make-bounded-counter init bottom top) (list init bottom top)) Selectors: (define (counter-value c) (car c)) (define (counter-bottom c) (cadr c)) (define (counter-top c) (caddr c))

12 List Implementation – cont. Mutators: (define (counter-inc! c) (if (< (counter-value c) (counter-top c)) (set-car! c (+ 1 (counter-value c)))) (counter-value c)) (define (counter-dec! c) (if (> (counter-value c) (counter-bottom c)) (set-car! c (- (counter-value c) 1))) (counter-value c))

13 Polish Notation PostFix Notation: operands before operators Expressions with binary operators can be written without Parentheses Apply operators, from left to right, on the last two numbers – * + –5 3 2 * + –5 6 + – 11 Stack implementation: –Init: Empty stack –Number: insert! Into stack –Operator: apply on 2 top stack elements and insert! result into stack

14 Simulation (calc) TOP= 3 2 * TOP= 6 + TOP= 11 exit ok

15 Read & Eval (read) - returns an expression from the user (eval exp) - evaluates an expression We will use these functions in: –(perform m) - receives a symbol of an operation, applies it on the two top numbers in the stack, and returns the result to the stack –(iter) - reads an input from the user. If it is a number it is pushed to the stack, if it is an operator we call perform

16 perform (define (perform m) (let ((arg2 ((stk 'top)))) ((stk 'delete!)) (let ((arg1 ((stk 'top)))) ((stk 'delete!)) ((stk 'insert!) ((eval m) arg1 arg2)))))

17 iter (define (iter) (let ((in (read))) (if (eq? in 'exit) 'ok (begin (cond ((number? in) ((stk 'insert!) in)) (else (perform in) (display "TOP= ") (display ((stk 'top))) (newline))) (iter)))))

18 Calc (define (calc) (let ((stk (make-stack))) (define (perform m)...) (define (iter)... ) (iter)))

19 Vectors Constructors: (vector v1 v2 v3...) (make-vector size init) Selector: (vector-ref vec place) Mutator: (vector-set! vec place value) Other functions: (vector-length vec)

20 Example - accumulating (define (accumulate-vec op base vec) (define (helper from to) (if (> from to) base (op (vector-ref vec from) (helper (+ from 1) to)))) (helper 0 (- (vector-length vec) 1)))

21 Bucket Sort Problem: Sorting numbers that distribute “uniformly” across a given interval (for example, [0,1) ). Observation: The number of elements that fall within a sub-interval is proportional to the sub-interval’s size Idea: –Divide into sub-intervals (buckets) –Throw each number into appropriate bucket –Sort within each bucket (any sorting method) –Adjoin all sorted buckets

22 Example If we want ~3 numbers in each bucket, we need 3-4 buckets. Using 3 buckets we have: Bucket : Bucket – 0.667: Bucket – 1 : Sorting: Now sort! Bucket : Bucket – 0.667: Bucket – 1 :

23 Analysis Observation: If number of buckets is proportional to the number of elements, then the number of elements in each bucket is more or less the same, and bounded by a constant. Dividing into buckets: O(n) Sorting one bucket: O(1) Sorting all buckets: O(n) Adjoining sorted sequences: O(n)

24 Implementation A bucket is a list The set of buckets is a vector The elements are sorted while inserted into the buckets (very similar so insertion sort): (define (insert x s) (cond ((null? s) (list x)) ((< x (car s)) (cons x s)) (else (cons (car s) (insert x (cdr s))))))

25 Implementation – for-each Scheme primitive Similar to map, without returning a value Useful for side-effects (define (for-each proc lst) (if (null? lst) ‘done (begin (proc (car lst)) (for-each proc (cdr lst)))))

26 Implementation – cont. (define (bucket-sort s) (let* ((size 5) (n (ceiling (/ (length s) size))) (buckets (make-vector n null))) (define (insert! x) (let ((bucket (inexact->exact (floor (* n x))))) (vector-set! buckets bucket (insert x (vector-ref buckets bucket))))) (for-each insert! s) (accumulate-vec append null buckets)))

Streams 3.5, pages definitions file on web 27

cons, car, cdr (define s (cons 9 (begin (display 7) 5))) -> prints 7 The display command is evaluated while evaluating the cons. (car s) -> 9 (cdr s) -> 5 28

cons-stream, stream-car, stream-cdr (define s (cons-stream 9 (begin (display 7) 5))) Due to the delay of the second argument, cons-stream does not activate the display command (stream-car s) -> 9 (stream-cdr s) -> prints 7 and returns 5 stream-cdr activates the display which prints 7, and then returns 5. 29

List enumerate (define (enumerate-interval low high) (if (> low high) nil (cons low (enumerate-interval (+ low 1) high)))) (enumerate-interval 2 8) -> ( ) (car (enumerate-interval 2 8)) -> 2 (cdr (enumerate-interval 2 8)) -> ( ) 30

Stream enumerate (define (stream-enumerate-interval low high) (if (> low high) the-empty-stream (cons-stream low (stream-enumerate-interval (+ low 1) high)))) (stream-enumerate-interval 2 8) -> (2. # ) (stream-car (stream-enumerate-interval 2 8)) -> 2 (stream-cdr (stream-enumerate-interval 2 8)) -> (3. # ) 31

List map (map ) (define (map proc s) (if (null? s) nil (cons (proc (car s)) (map proc (cdr s))))) (map square (enumerate-interval 2 8)) -> ( ) 32

Stream map (map ) (define (stream-map proc s) (if (stream-null? s) the-empty-stream (cons-stream (proc (stream-car s)) (stream-map proc (stream-cdr s)) ))) (stream-map square (stream-enumerate-interval 2 8)) -> (4. # ) 33

List of squares (define squares (map square (enumerate-interval 2 8))) squares -> ( ) (car squares) -> 4 (cdr squares) -> ( ) 34

Stream of squares (define stream-squares (stream-map square (stream-enumerate-interval 2 8))) stream-squares -> (4. # ) (stream-car stream-squares) -> 4 (stream-cdr stream-squares) -> (9. # ) 35

List reference (define (list-ref s n) (if (= n 0) (car s) (list-ref (cdr s) (- n 1)))) (define squares (map square (enumerate-interval 2 8))) (list-ref squares 3) -> 25 36

Stream reference (define (stream-ref s n) (if (= n 0) (stream-car s) (stream-ref (stream-cdr s) (- n 1)))) (define stream-squares (stream-map square (stream-enumerate-interval 2 8))) (stream-ref stream-squares 3) -> 25 37

List filter (filter ) (define (filter pred s) (cond ((null? s) nil) ((pred (car s)) (cons (car s) (filter pred (cdr s)))) (else (filter pred (cdr s))))) (filter even? (enumerate-interval 1 20)) -> ( ) 38

Stream filter (stream-filter ) (define (stream-filter pred s) (cond ((stream-null? s) the-empty-stream) ((pred (stream-car s)) (cons-stream (stream-car s) (stream-filter pred (stream-cdr s)))) (else (stream-filter pred (stream-cdr s))) ))) (stream-filter even? (stream-enumerate-interval 1 20)) -> (2. # ) 39

Generalized list map (generalized-map … ) (define (generalized-map proc. arglists) (if (null? (car arglists)) nil (cons (apply proc (map car arglists)) (apply generalized-map (cons proc (map cdr arglists)))))) (generalized-map + squares squares squares) -> ( ) 40

Generalized stream map (generalized-stream-map … ) (define (generalized-stream-map proc. argstreams) (if (stream-null? (car argstreams)) the-empty-stream (cons-stream (apply proc (map stream-car argstreams)) (apply generalized-stream-map (cons proc (map stream-cdr argstreams)))))) (generalized-stream-map + stream-squares stream-squares stream-squares) -> (12. # ) 41

List for each (define (for-each proc s) (if (null? s) 'done (begin (proc (car s)) (for-each proc (cdr s))))) 42

Stream for each (define (stream-for-each proc s) (if (stream-null? s) 'done (begin (proc (stream-car s)) (stream-for-each proc (stream-cdr s))))) useful for viewing (finite!) streams (define (display-stream s) (stream-for-each display s)) (display-stream (stream-enumerate-interval 1 20)) -> prints 1 … 20 done 43

Lists (define sum 0) (define (acc x) (set! sum (+ x sum)) sum) (define s (map acc (enumerate-interval 1 20))) s -> ( ) sum -> 210 (define y (filter even? s)) y -> ( ) sum -> 210 (define z (filter (lambda (x) (= (remainder x 5) 0)) s)) z -> ( ) sum ->

(list-ref y 7) -> 136 sum -> 210 (display z) -> prints ( ) sum ->

Streams (define sum 0) (define (acc x) (set! sum (+ x sum)) sum) (define s (stream-map acc (stream-enumerate-interval 1 20))) s -> (1. # )sum -> 1 (define y (stream-filter even? s)) y -> (6. # )sum -> 6 (define z (stream-filter (lambda (x) (= (remainder x 5) 0)) s)) z -> (10. # )sum -> 10 46

(stream-ref y 7) -> 136 sum -> 136 (display-stream z) -> prints done sum ->

Defining streams implicitly by delayed evaluation Suppose we needed an infinite list of Dollars. We can (define bill-gates (cons-stream ‘dollar bill-gates)) If we need a Dollar we can take the car (stream-car bill-gates) -> dollar The cdr would still be an infinite list of Dollars. (stream-cdr bill-gates)->(dollar. # ) 48

49 Infinite Streams Formulate rules defining infinite series wishful thinking is key 49

1,ones (define ones (cons-stream 1 ones)) 1,1,1,… = ones = 50

2,twos (define twos (cons-stream 2 twos)) ones + ones adding two infinite series of ones (define twos (stream-map + ones ones)) 2 * ones element-wise operations on an infinite series of ones (define twos (stream-map (lambda (x) (* 2 x)) ones)) or (+ x x) 2,2,2,… = twos = 51

1,2,3,… = integers = 1,ones + integers 1,1,1… 1,2,3,… 2,3,4,… (define integers (cons-stream 1 (stream-map + ones integers))) + 52

0,1,1,2,3,… = fibs = 0,1,fibs + (fibs from 2 nd position) 0,1,1,2,… 1,1,2,3,… 1,2,3,5,… (define fibs (cons-stream 0 (cons-stream 1 (stream-map + fibs (stream-cdr fibs))))) + 53

1,doubles + doubles 1,2,4,8,… 2,4,8,16,… (define doubles (cons-stream 1 (stream-map + doubles doubles))) 1,2,4,8,… = doubles = + 54

1,2 * doubles (define doubles (cons-stream 1 (stream-map (lambda (x) (* 2 x)) doubles))) or (+ x x) 1,2,4,8,… = doubles = 55

1,factorials * integers from 2 nd position 1, 1*2, 1*2*3,… 2, 3, 4,… 1*2,1*2*3,1*2*3*4,… (define factorials (cons-stream 1 (stream-map * factorials (stream-cdr integers)))) 1,1x2,1x2x3,... = factorials = x 56

(1),(1 2),(1 2 3),… = runs = (1), append runs with a list of integers from 2 nd position (1), (1 2), (1 2 3),… (2), (3), (4),… (1 2),(1 2 3),( ),… (define runs (cons-stream (list 1) (stream-map append runs (stream-map list (stream-cdr integers))))) append 57

a0,a0+a1,a0+a1+a2,… = partial sums = a0,partial sums + (stream from 2 nd pos) a0, a0+a1, a0+a1+a2,… a1, a2, a3,… a0+a1,a0+a1+a2,a0+a1+a2+a3,… (define (partial-sums a) (cons-stream (stream-car a) (stream-map + (partial-sums a) (stream-cdr a)))) + 58

59 Partial Sums (cont.) (define (partial-sums a) (define sums (cons-stream (stream-car a) (stream-map + sums (stream-cdr a)))) sums) This implementation is more efficient since it uses the stream itself rather than recreating it recursively 59

Repeat Input: procedure f of one argument, number of repetitions Output: f*…*f, n times (define (repeated f n) (if (= n 1) f (compose f (repeated f (- n 1))))) (define (compose f g) (lambda (x) (f (g x)))) 60

Repeat stream f,f*f,f*f*f,… = repeat = f,compose f,f,f,… with repeat (define f-series (cons-stream f f-series)) (define stream-repeat (cons-stream f (stream-map compose f-series stream-repeat))) We would like f to be a parameter 61

Repeat stream f,f*f,f*f*f,… = repeat = f,compose f,f,f,… with repeat (define (make-repeated f) (define f-series (cons-stream f f-series)) (define stream-repeat (cons-stream f (stream-map compose f-series stream-repeat))) stream-repeat) 62

Interleave 1,1,1,2,1,3,1,4,1,5,1,6,… (interleave ones integers) s0,t0,s1,t1,s2,t2,… interleave = s0,interleave (t, s from 2 nd position) (define (interleave s t) (if (stream-null? s) t (cons-stream (stream-car s) (interleave t (stream-cdr s))))) 63