Representing lists as binary trees

1. 3 1 7 3 7 1 9 3 5 9 5 12 5 1 12 7 9 12 Representing lists as binary trees מבוא מורחב

2. 7 9 3 12 5 1 Representing lists as binary trees (Cont.) (define (entry tree) (car tree)) (define (left-branch tree) (cadr tree)) (define (right-branch tree) (caddr tree)) (define (make-tree entry left right) (list entry left right)) מבוא מורחב

3. Representing lists as binary trees (Cont.) (define (adjoin-set x set) (cond ((null? set) (make-tree x '() '())) ((= x (entry set)) set) ((< x (entry set)) (make-tree (entry set) (adjoin-set x (left-branch set)) (right-branch set))) ((> x (entry set)) (make-tree (entry set) (left-branch set) (adjoin-set x (right-branch set)))))) מבוא מורחב

4. unordered ordered Element-of-set Adjoin-set Intersection-set Union-set (n) (n) (n) (n) (n) (n2) (n) (n2) Complexity trees (log(n)) (log(n)) (nlog(n)) (nlog(n)) מבוא מורחב

5. Huffman encoding trees מבוא מורחב

6. fixed length codes Want to represent data as a sequence of 0’s and 1’s for example: BACADAEAFABBAAAGAH A 000 B 001 C 010 D 011 E 100 F 101 G 110 H 111 001000010000011000100000101000001001000000000110000111 This is a fixed length code. Can we make the sequence of 0’s and 1’s shorter ? מבוא מורחב

7. variable length prefix codes A 0 B 100 C 1010 D 1011 E 1100 F 1101 G 1110 H 1111 100010100101101100011010100100000111001111 This is a variable length code. How do we decode ? Use prefix codes: No codeword is a prefix of the other מבוא מורחב

8. A B C D E F G H prefix code <==> binary tree A 0 B 100 C 1010 D 1011 E 1100 F 1101 G 1110 H 1111 A prefix code corresponds to a binary tree with the symbols at the leaves and vice versa.

9. Representation {A,B,C,D,E,F,G,H} 17 A 8 9 {B,C,D,E,F,G,H} 4 5 {B,C,D} {E,F,G,H} 2 2 B 2 {C,D} {G,H} {E,F} 3 C D E F G H 1 1 1 1 1 1

10. Representation (Cont.) (define (make-leaf symbol weight) (list 'leaf symbol weight)) (define (leaf? object) (eq? (car object) 'leaf)) (define (symbol-leaf x) (cadr x)) (define (weight-leaf x) (caddr x)) מבוא מורחב

11. Representation (Cont.) (define (make-code-tree left right) (list left right (append (symbols left) (symbols right)) (+ (weight left) (weight right)))) (define (left-branch tree) (car tree)) (define (right-branch tree) (cadr tree)) מבוא מורחב

12. Representation (Cont.) (define (symbols tree) (if (leaf? tree) (list (symbol-leaf tree)) (caddr tree))) (define (weight tree) (if (leaf? tree) (weight-leaf tree) (cadddr tree))) מבוא מורחב

13. {A,B,C,D,E,F,G,H} 17 9 {B,C,D,E,F,G,H} 4 {E,F,G,H} 5 {B,C,D} 2 2 2 {C,D} {G,H} {E,F} C D E F G B A H 1 1 1 1 3 1 1 8 Construction of Huffman tree

14. Construction of Huffman tree (generate-huffman-tree '((A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (H 1) (G 1)) ((leaf a 8) ((((leaf g 1) (leaf h 1) (g h) 2) ((leaf f 1) (leaf e 1) (f e) 2) (g h f e) 4) (((leaf d 1) (leaf c 1) (d c) 2) (leaf b 3) (d c b) 5) (g h f e d c b) 9) (a g h f e d c b) 17)

15. Construction of Huffman tree (define (generate-huffman-tree pairs) (successive-merge (make-leaf-set pairs))) (define (make-leaf-set pairs) (if (null? pairs) '() (let ((pair (car pairs))) (adjoin-set (make-leaf (car pair) (cadr pair)) (make-leaf-set (cdr pairs))))))

16. Construction of Huffman tree (define (adjoin-set x set) (cond ((null? set) (list x)) ((< (weight x) (weight (car set))) (cons x set)) (else (cons (car set) (adjoin-set x (cdr set)))))) (define (successive-merge trees) (if (null? (cdr trees)) (car trees) (let ((smallest (car trees)) (2smallest (cadr trees)) (rest (cddr trees))) (successive-merge (adjoin-set (make-code-tree smallest 2smallest) rest)))))

17. Huffman’s encoding trees (Cont.) (define (decode bits tree) (define (decode-1 bits current-branch) (if (null? bits) '() (let ((next-branch (choose-branch (car bits) current-branch))) (if (leaf? next-branch) (cons (symbol-leaf next-branch) (decode-1 (cdr bits) tree)) (decode-1 (cdr bits) next-branch))))) (decode-1 bits tree)) (define (choose-branch bit branch) (cond ((= bit 0) (left-branch branch)) ((= bit 1) (right-branch branch)) (else (error "bad bit -- CHOOSE-BRANCH" bit)))) מבוא מורחב

18. {A,B,C,D,E,F,G,H} 17 9 {B,C,D,E,F,G,H} 4 {E,F,G,H} 5 {B,C,D} 2 2 2 {C,D} {G,H} {E,F} C D E F G B A H 1 1 1 1 3 1 1 8 Decoding example 10001010

19. Midterm last spring (cont) השלם/י את מימוש השגרה (max-ind lst-ind) שמקבלת רשימה שהתקבלה מהפעלתadd-ind על רשימת מספרים, ומחזירה את הזוג שבו האיבר השני הוא הגדול ביותר. לדוגמא: (max-ind (add-ind 0 (list 10 17 2 5))) ==> (1 . 17) (define (max-ind lst-ind) (accumulate ________________________________ ____________ ____________ )) (lambda (x y) (if (> (cdr x) (cdr y)) x y)) (car lst-ind) lst-ind מבוא מורחב

20. Midterm last spring (Cont) תהאlst רשימת מספרים שכל איבריה שונים. הגדר שגרה (break-list lst) שמחזירה רשימה באורך שלוש. האיבר הראשון ברשימה שמוחזרת היא רשימת איבריlst שמופיעים לפני האיבר המירבי שלlst. האיבר השני ברשימה הוא האיבר המירבי ב-lst. האיבר השלישי הוא רשימת איבריlst שמופיעים לאחר האיבר המירבי. לדוגמא: (break-list ‘(1 8 3 6 4 9 2 5 7))  ( (1 8 3 6 4) 9 (2 5 7) ) (break-list ‘(5 4 3 2 1))  ( () 5 (4 3 2 1) ) (break-list ‘(7))  ( () 7 () ) מבוא מורחב