1

Append: process

(append list1 list2) (cons 1 (append ‘(2) list2)) (cons 1 (cons 2 (append ‘() list2))) (cons 1 (cons 2 list2))

(define (append list1 list2) (cond ((null? list1) list2) ; base (else (cons (car list1) ; recursion (append (cdr list1) list2)))))

1 2 3 4

list2list1

1 2

(1 2 3 4)

(define list1 (list 1 2))(define list2 (list 3 4))

Quote: what you see is what you get.

04/20/23 2

Reverse of a list

(reverse (list 1 2 3 4))

(define (reverse lst)(cond ((null? lst) lst)(else ( (reverse (cdr lst))

)))))append

(list (car lst))

(append )

(reverse (cdr lst)) (list (car lst))

4 3 2 1

4 3 2 1

Append: T(n) = c*n = (n)

Reverse: T(n) = c*(n-1) + c*(n-2)+

… + c*1 = (n2)

4

Trees

2 4

6 8

Abstract tree:

• a leaf (a node that has no children, and contains data) - is a tree

• an internal node (a node whose children are trees) – is a tree leaf

leaf

86

2 4

internal node

internal node

Implementation of tree:

• a leaf - will be the data itself

• an internal node – will be a list of its children

5

Count Leaves of a Tree

• Strategy– base case: count of an empty tree is 0– base case: count of a leaf is 1– recursive strategy: the count of a tree is the sum of the countleaves of each child in the tree.

• Implementation:

(define (leaf? x) (atom? x))

6

Count Leaves(define (countleaves tree) (cond ((null? tree) 0) ;base case ((leaf? tree) 1) ;base case (else ;recursive case (+ (countleaves (car tree)) (countleaves (cdr tree))))))

(define my-tree

(list 4 (list 5 7) 2))4 2

5 7

7

Countleaves

(countleaves my-tree ) ==> 4

(cl (4 (5 7) 2))

+

(cl 4) (cl ((5 7) 2) )

+

(cl (5 7)) (cl (2))+

(cl 2) (cl null)

+

(cl 5) (cl (7))+

(cl 7) (cl null)

1

1

1 0

1 0

my-tree

4 2

5 7

1

12

3

4

8

Enumerate-Leaves

• Goal: given a tree, produce a list of all the leaves • Strategy

– base case: list of empty tree is empty list– base case: list of a leaf is one element list– otherwise, recursive strategy: build a new list from

a list of the leaves of the first child and a list of the leaves of the rest of the children

9

Enumerate-Leaves(define (enumerate-leaves tree) (cond ((null? tree) null) ;base case ((leaf? tree) ) ;base case (else ;recursive case ( (enumerate-leaves (car tree)) (enumerate-leaves (cdr tree))))))

10

Enumerate-Leaves(define (enumerate-leaves tree) (cond ((null? tree) null) ;base case ((leaf? tree) (list tree)) ;base case (else ;recursive case (append (enumerate-leaves (car tree)) (enumerate-leaves (cdr tree))))))

4 25 7

4 2

5 7

8מבוא מורחב - שיעור 11

Enumerate-leaves

(el (4 (5 7) 2))

ap

(el 4) (el ((5 7) 2) )

ap

(cl (5 7)) (el (2))ap

(el 2) (el nil)

ap

(el 5) (el (7))ap

(el 7) (el nil)

(4)

(5)

(7) ()

(2) ()

(7)

(5 7) (2)

(5 7 2)

(4 5 7 2)

8מבוא מורחב - שיעור 12

Your Turn: Scale-tree

• Goal: given a tree, produce a new tree with all the leaves scaled

• Strategy– base case: scale of empty tree is empty tree– base case: scale of a leaf is product– otherwise, recursive strategy: build a new tree

from a scaled version of the first child and a scaled version of the rest of children

13

Scale-tree

(define (scale-tree tree factor)

(cond ((null? tree) ) ;base case

((leaf? tree) )

(else ;recursive case

(cons

))))

(scale-tree (car tree) factor)

null

(* tree factor)

(scale-tree (cdr tree) factor)

14

Enumeration

)integers-between 2 4()cons 2 (integers-between 3 4)((

)cons 2 (cons 3 (integers-between 4 4))()cons 2 (cons 3 (cons 4 (integers-between 5 4)))(

)cons 2 (cons 3 (cons 4 null))()2 3 4(

(define (integers-between lo hi) (cond ((> lo hi) null) (else (cons lo (integers-between (+ 1 lo) hi)))))

2 3 4

15

Count Leaves(define (countleaves tree) (cond ((null? tree) 0) ;base case ((leaf? tree) 1) ;base case (else ;recursive case (+ (countleaves (car tree)) (countleaves (cdr tree))))))

(define my-tree

(list 4 (list 5 7) 2))4 2

5 7

16

List abstraction (sequence operations)

• Find common high order patterns• Distill them into high order procedures• Use these procedures to simplify list operations

• Mapping• Filtering• Accumulating

Patterns:

17

Mapping

(define (map proc lst) (if (null? lst) null (cons (proc (car lst)) (map proc (cdr lst)))))

(define (square-list lst) (map square lst))

(define (scale-list lst c) (map (lambda (x) (* c x)) lst))

(scale-list (integers-between 1 5) 10) ==>(10 20 30 40 50)

18

Mapping: process(define (map proc lst) (if (null? lst) null (cons (proc (car lst)) (map proc (cdr lst)))))

(map square (list 1 2 3))(cons (square 1) (map square (list 2 3)))(cons 1 (map square (list 2 3)))(cons 1 (cons (square 2) (map square (list 3))))(cons 1 (cons 4 (map square (list 3))))(cons 1 (cons 4 (cons (square 3) (map square null))))(cons 1 (cons 4 (cons 9 (map square null))))(cons 1 (cons 4 (cons 9 null)))(1 4 9)

19

Generalized Mapping

(map + (list 1 2 3) (list 10 20 30) (list 100 200 300))

==> (111 222 333)

(map (lambda (x y) (+ x (* 2 y))) (list 1 2 3) (list 4 5 6))

==> (9 12 15)

(map <proc> <list1>…<listn>)

Returns a list in which proc is applied to the i-th elements of the lists respectively.

We will see how to write such a procedure later!

20

Filtering

(define (filter pred lst)( cond ((null? lst) null)

(( pred (car lst))( cons (car lst)

( filter pred (cdr lst))))( else (filter pred (cdr lst)))))

(filter odd? (integers-between 1 10))(1 3 5 7 9)

21

Filtering: process(define (filter pred lst)

( cond ((null? lst) null)(( pred (car lst))

( cons (car lst)( filter pred (cdr lst))))

( else (filter pred (cdr lst)))))

(filter odd? (list 1 2 3 4))(cons 1 (filter odd? (list 2 3 4)))(cons 1 (filter odd? (list 3 4)))(cons 1 (cons 3 (filter odd? (list 4))))(cons 1 (cons 3 (filter odd? null)))(cons 1 (cons 3 null))(1 3)

22

Finding all the Primes: Sieve of Eratosthenes (a.k.a. Beta)

XX XX

XX

XX

XX XX

XX

XX XX

XX

XX XX

XX

7

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XXXX X 2

XX XX XX

XX XX XX XX

XX XX XX

XX XX XX

XX XX XX XX

XX XX XX

XX XX XX

XX XX XX XX

XX XX XX

X X 3

100999897969594939291

90898887868584838281

80797877767574737271

60696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

1098765432

XX XX

XX XX

XX XX

XX XX

XX XX

XX XX

XX XX

XX XX

XX XX

XX 5

23

..And here’s how to do it!(define (sieve lst)

( if (null? lst)

)(‘

( cons (car lst)

( sieve (filter

( lambda (x)

( not (divisible? x (car lst))))

( cdr lst))))))

(cons 2 (sieve (filter (lambda (x) (not (divisible? X 2) (list 3 4 5 …100 ))))

==> (sieve (list 2 3 4 5 … 100))

24

How sieve works

• Sieve takes as argument a list of numbers L and returns a list M.

• Take x, the first element of L and make it the first element in M.

• Drop all numbers divisible by x from (cdr L).

• Call sieve on the resulting list, to generate the rest of M.

(define (sieve lst)

(if (null? lst)

‘()

(cons (car lst)

(sieve (filter

(lambda (x)

(not (divisible? x (car lst))))

(cdr lst))))))

25

Another example

Find the number of integers x in the range [1…100] s.t:.

x * (x + 1) is divisible by 6.

(length

(filter _____________________________________

(map _________________________________

_________________________________))))

(lambda(n) (* n (+ n 1)))

(integers-between 1 100)

(lambda(n) (= 0 (remainder n 6)))

Any bets on the result???? 66 (about two thirds)

26

A new pattern: Accumulation

27

Accumulating(define (add-up lst)

( if (null? lst) 0

( +( car lst)( add-up (cdr lst)))))

(define (mult-all lst) (if (null? lst) 1 (* (car lst) (mult-all (cdr lst)))))

(define (accumulate op init lst) (if (null? lst) init (op (car lst) (accumulate op init (cdr lst)))))

Add up the elements of a list

Multiply all the elements of a list

28

Accumulating (cont.)(define (accumulate op init lst) (if (null? lst) init (op (car lst) (accumulate op init (cdr lst)))))

(define (add-up lst) (accumulate + 0 lst))(define (mult-all lst) (accumulate * 1 lst))

eln init

op

eln-1 el1 ……..

opop

op...

29

Accumulate: process(define (accumulate op init lst) (if (null? lst) init (op (car lst) (accumulate op init (cdr lst)))))

(accumulate append ‘()(list (list 1) (list 2) (list 3))) (append ‘(1) (accumulate append ‘() ‘((2)(3)))) (append ‘(1) (append ‘(2) (accumulate append ‘()‘((3))))) (append ‘(1) (append ‘(2) (append ‘(3) (accumulate append ‘()‘())))) (append ‘(1) (append ‘(2) (append ‘(3) ‘()))) (append ‘(1) (append ‘(2) ‘(3))) (append ‘(1) ‘(2 3))

(1 2 3)

1 2 3

31 2

30

Length and append as accumulation

(define (length lst) (accumulate (lambda (x y) (+ 1 y)) 0 lst))

(define (append lst1 lst2) (accumulate cons lst2 lst1)

0eln

+1

eln-1 el1 ……..

+1+1

+1...

eln lst2eln-1 el1 ……..

...

31

Implementing map and filterusing accumulate

(define (map proc lst) (accumulate (lambda (x y) (cons (proc x) y)) ‘() lst))

(define (filter pred lst) (accumulate (lambda (x y) (if (pred x) (cons x y) y) ‘() lst))

32

Another accumulate example

We need to implement (more-evens lst), that receives a list of integers and returns #t iff lst contains more even integers than odd integers.

(define (more-evens? l)

(positive? (accumulate _________________________ _________________________ ___ l)))

(lambda(x y) (+ y (if (even? x) 1 –1)))

0

33

Yet another example: dot product

(dot-product x y) returns i xiyi

(define (dot-product x y)

(accumulate ) + 0 (map * x y)

The generalized map!

04/20/23 34

Lists as interfaces

04/20/23 35

(define (sum-odd-squares tree) (cond ((null? tree) 0) ((leaf? tree) (if (odd? tree) (square tree) 0)) (else (+ (sum-odd-squares (car tree)) (sum-odd-squares (cdr tree))))))

(define (even-fibs n) (define (next k) (if (< n k)

nil (let ((f (fib k))) (if (even? f) (cons f (next (+ k 1))) (next (+ k 1)))))) (next 0))

Common Structure

04/20/23 36

Even-fibs

• Enumerates the integers from 0 to n

• Computes the Fibonacci number for each integer

• Filters them selecting the even ones

• Accumulates the results using cons

Sum-odd-squares

• Enumerates the leaves of a tree

• Filters them, selecting the odd ones

• Squares each of the selected ones

• Accumulates the results using +

04/20/23 37

enumerate leaves

filter map accumulate

treeodd? square + 0

integersbetween

map filter accumulate

0, nfib even? cons nil

even-fibs

sum-odd-squares in a tree

Interface: Along horizontal arrows flow lists of numbers

04/20/23 38

even-fibs - Implementation

(define (even-fibs n) (accumulate cons nil (filter even? (map fib

(integers-between 0 n)))))

integersbetween

map filter accumulate

0, nfib even? cons nil

sum-odd-squares - Implementation

(define (sum-odd-squares tree) (accumulate + 0 (map square (filter odd? (enumerate-leaves tree)))))

enumerate leaves

filter map accumulate

treeodd? square + 0

40

Example : Even-fibs-prod (product of all fib numbers with even index between lo and hi)

• Enumerate the integers from lo to hi

• Filter them selecting the even ones

• Compute the Fibonacci number for each integer using map

• Accumulates the results using * and 1

Lists as interfaces• An algorithm as a series of steps performed on sequences (lists).

• Each step performs a simple operation.

• Start with a given list, or by creating a list.

• Other steps use filter or map.

• Finally, generate final result by accumulation.

41

(define (even-fibs-prod lo hi) (accumulate * 1 (map fib (filter even? (integers-between lo hi)))))

Compare with: (define (even-fibs-prod lo hi) (cond ((> lo hi) 1) ((even? lo) (* (fib lo) (even-fibs-prod (+ lo 1) hi))) (else (even-fibs-prod (+ lo 1) hi))))

even-fibs-prod

42

Even-fibs-prod

Interface: Along horizontal arrows flow lists of numbers

generate filter map accumulate

lo, hieven? fib * 0

integers between

generate filter/map filter/map accumulate....

(optional)

A common way to structure a computation:

43

What did we gain

• A conventional way to partition a problem into small tasks.

• Resulting code is clearer, easier to design and understand.

• But sometimes less efficient.