Lecture #10

1. Lecture #10 Sets (2.3.3) Huffman Encoding Trees (2.3.4) מבוא מורחב

2. Sets A set is a collection of distinct items. Methods: (element-of-set? x set) (adjoin-set x set) (union-set s1 s2) (intersection-set s1 s2) מבוא מורחב

3. Implementation • We will see three ways to implement sets. • With lists • With sorted lists • With trees • And compare the three methods. מבוא מורחב

4. First implementation: Lists Empty set  empty list ‘() Adding an element  cons Set Union  append Set Intersection  But: we also need to remember to remove duplicats. מבוא מורחב

5. First implementation: Lists (define (element-of-set? x set) (cond ((null? set) false) ((equal? x (car set)) true) (else (element-of-set? x (cdr set))))) equal? : Like eq? for symbols. Works for numbers Works recursively for compounds. (eq? (list ‘a ‘b) (list ‘a ‘b)) #f (equal? (list ‘a ‘b) (list ‘a ‘b)) #t מבוא מורחב

6. First Implementation: Adjoin (define (adjoin-set x set) (if (element-of-set? x set) set (cons x set))) מבוא מורחב

7. Intersection (define (intersection-set set1 set2) (cond ((or (null? set1) (null? set2)) '()) ((element-of-set? (car set1) set2) (cons (car set1) (intersection-set (cdr set1) set2))) (else (intersection-set (cdr set1) set2)))) מבוא מורחב

8. Union (define (union-set set1 set2) (cond ((null? set1) set2)) ((not (element-of-set? (car set1) set2)) (cons (car set1) (union-set (cdr set1) set2))) (else (union-set (cdr set1) set2)))) (define (union-set set1 set2) (cond ((null? set1) set2)) (else (adjoin-set (car set1) (union-set (cdr set1) set2))))) מבוא מורחב

9. Analysis (n) Element-of-set Adjoin-set Intersection-set Union-set (n) (n2) (n2) מבוא מורחב

10. Second implementation: Sorted Lists Empty set  empty list ‘() Adding an element  We add the element to the list so that the list is sorted. Set Union  sorted list of union. Set Intersection  sorted list of intersection. מבוא מורחב

11. Membership? (define (element-of-set? x set) (cond ((null? set) false) ((= x (car set)) true) ((< x (car set)) false) (else (element-of-set? x (cdr set))))) (n) steps. adjoin-set is similar, try it yourself מבוא מורחב

12. Intersection (define (intersection-set set1 set2) (cond ((or (null? set1) (null? set2)) '()) ((element-of-set? (car set1) set2) (cons (car set1) (intersection-set (cdr set1) set2))) (else (intersection-set (cdr set1) set2)))) Can we do it better ? מבוא מורחב

13. Better Intersection (define (intersection-set set1 set2) (if (or (null? set1) (null? set2)) '() (let ((x1 (car set1)) (x2 (car set2))) (cond ((= x1 x2) (cons x1 (intersection-set (cdr set1) (cdr set2)))) ((< x1 x2) (intersection-set (cdr set1) set2)) ((< x2 x1) (intersection-set set1 (cdr set2))))))) מבוא מורחב

14. Example set1 set2 intersection (1 3 7 9) (1 4 6 7) (1 (3 7 9) (4 6 7) (1 (7 9) (4 6 7) (1 (7 9) (6 7) (1 (7 9) (7) (1 (9) () (1 7) Time and space  (n) Union -- similar מבוא מורחב

15. unsorted Element-of-set Adjoin-set Intersection-set Union-set (n) (n) (n2) (n2) Complexity sorted (n) (n) (n) (n) מבוא מורחב

16. 3 7 7 1 9 3 5 9 12 5 1 12 Version 3: Binary Trees Depth = (log n) Balanced Tree Unbalanced Tree מבוא מורחב

17. 7 9 3 12 5 1 Interface of Binary Trees. Constructor: (define (make-tree entry left right) (list entry left right)) Selectors: (define (entry tree) (car tree)) (define (left-branch tree) (cadr tree)) (define (right-branch tree) (caddr tree)) מבוא מורחב

18. Element-of-set (define (element-of-set? x set) (cond ((null? set) false) ((= x (entry set)) true) ((< x (entry set)) (element-of-set? x (left-branch set))) ((> x (entry set)) (element-of-set? x (right-branch set))))) Complexity: (d) If tree is balanced d  log(n) מבוא מורחב

19. unsorted sorted Element-of-set Adjoin-set Intersection-set Union-set (n) (n) (n) (n) (n) (n2) (n) (n2) Complexity trees (log(n)) (log(n)) (n log(n)) (n log(n)) מבוא מורחב

20. Huffman encoding trees מבוא מורחב

21. Data Transmission “sos” A B We wish to send information efficiently from A to B מבוא מורחב

22. Fixed Length Codes Represent data as a sequence of 0’s and 1’s Example: BACADAEAFABBAAAGAH A000 B 001 C 010 D 011 E 100 F 101 G 110 H 111 001000010000011000100000101000001001000000000110 000111 This is a fixed length code. Sequence is 18x3=54 bits long. Can we make the sequence of 0’s and 1’s shorter ? מבוא מורחב

23. 42 bits (20% shorter) Variable Length Code Make use of frequencies. Frequency of A=8, B=3, others1. A 0 B 100 C 1010 D 1011 E 1100 F 1101 G 1110 H 1111 Example: BACADAEAFABBAAAGAH 100010100101101100011010100100000111001111 But how do we decode? מבוא מורחב

24. 0 1 0 1 A 0 1 0 1 0 1 1 0 0 1 B C D E F G H Prefix code  Binary tree Prefix code: No codeword is a prefix of the other A 0 B 100 C 1010 D 1011 E 1100 F 1101 G 1110 H 1111

25. 0 1 0 1 A 0 1 0 1 0 1 1 0 0 1 B C D E F G H Decoding Example 10001010 10001010 B 10001010 BA 10001010 BAC

26. Decoding a Message (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

27. Decoding a Message (define (decode bits Huffman_tree current_branch) (if (null? bits) '() (let ((next-branch (choose-branch (car bits) current_branch))) (if (leaf? next-branch) (cons (symbol-leaf next-branch) (decode (cdr bits) Huffman_tree Huffman_tree)) (decode (cdr bits) Huffman_tree next-branch))))) מבוא מורחב

28. The cost of a weighted Tree {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 3 {C,D} {E,F} {G,H} C D E F G H 1 1 1 1 1 1

29. 0 1 0 1 8 A 8 0 1 0 1 A 1 0 1 0 0 1 0 1 D 0 1 3 1 0 0 1 0 1 1 0 1 0 1 B C E F G H 1 3 1 1 1 1 B 1 1 1 1 1 1 C D E F G H Huffman Tree = Optimal Length Code Optimal: no code has better weighted average length

30. Next Time • We will describe an efficient algorithm that given the • weights, constructs the Huffman tree. • We will describe the interface it requires • and how to implement it. מבוא מורחב