1. 1 Assignment 2: (Due at 10:30 a.m on Friday of Week 10) Question 1 (Given in Tutorial 5) Question 2 (Given in Tutorial 7) If you do Question 1 only, you get 60 points. If you do Question 2 only, you get 90 points. If you correctly do both Question 1 and Question 2, you get 100 points. Bonus: 5 Points will be given to those who write a Java program for the Huffman code algorithm.

2. 2 Review of Lecture 1 to Lecture 6 Lecture 1: Some concept: Pseudo code, Abstract Data Type. (Page 60 of text book.) Stack. Give the ADT of stack (slide 11 of lecture1) The interface is on slide 19. (Q: Is the interface equivalent to ADT? Not really. We need the method for insertion and deletion, i.e., first in last out. ) Applications: parentheses matching

3. 3 Lecture 2: Linked list Singly linked list Doubly linked list Just know how to setup a list. (Assignment 1) Lecture 3: Analysis of Algorithms (important) Primitive operations Count number of primitive operations for an algorithm big-O notation 2n  O(n), 5n 2 +10n+11++>O(n 2 ).

4. 4 Lecture 4: Tree Definition of tree (slide 7) Tree terminology: root, internal node, external node (leaf), depth of a node, height of a node, height of a node. Inorder traversal of a binary tree Tree ADT, slide 11, Binary tree ADT, slide 17 In terms of programming, understand TreeInExample1.java. (If tested in exam, java codes will be given. I do not want to give long code.)

5. 5 Lecture 5: More on Trees Linked Structure for Binary Tree. Just understand the node: Preorder traversal for any tree Postorder traversal for any tree Array-Based representation of binary tree (slide 9) Algorithms for Depth(), Height() slide 12-15. 

6. 6 Lecture 6: Priority Queue (Heeps) Priority Queue ADT (slide 2) Heap: 1. definition of heap 2.What does “heap-order” mean? 3.Complete Binary tree (what is a complete binary?) 4.Height of a complete binary tree with n nodes is O(log n). 5.Insert a node into a heap runtimg time O(log n). 6.removeMin: remove a node with minimum key. Running time O(log n) Array-based complete binary tree representation. Show a sample exam paper.

7. 7 Lecture 6: Priority Queue (Heeps) Priority Queue ADT (slide 2) Heap: 1. definition of heap 2.What does “heap-order” mean? 3.Complete Binary tree (what is a complete binary?) 4.Height of a complete binary tree with n nodes is O(log n). 5.Insert a node into a heap runtimg time O(log n). 6.removeMin: remove a node with minimum key. Running time O(log n) Array-based complete binary tree representation. Show a sample exam paper.

8. 8 Exercise: Give some trees and ask students to give InOrder, PostOrder and PreOrder. Tutorial 6 of Question 2: Using PreOrder. Given a complete binary, write the array representation. Given an array, draw the complete binary tree. Given a heap, show the steps to removMin. Given a heap, show the steps to insert a node with key 3. (Do it for the tree version, do it for an array version.) Linear time construction of a heap.

9. Hash Tables9 Huffman codes (Page 565 Chapter 12.4) Binary character code: each character is represented by a unique binary string. A data file can be coded in two ways: abcdef frequency(%)4513121695 fixed-length code000001010011100101 variable-length code 010110011111011100 The first way needs 100  3=300 bits. The second way needs 45  1+13  3+12  3+16  3+9  4+5  4=232 bits.

10. Hash Tables10 Variable-length code Need some care to read the code. 001011101 (codeword: a=0, b=00, c=01, d=11.) Where to cut? 00 can be explained as either aa or b. Prefix of 0011: 0, 00, 001, and 0011. Prefix codes: no codeword is a prefix of some other codeword. (prefix free) Prefix codes are simple to encode and decode.

11. Hash Tables11 Using codeword in Table to encode and decode Encode: abc = 0.101.100 = 0101100 (just concatenate the codewords.) Decode: 001011101 = 0.0.101.1101 = aabe abcdef frequency(%)4513121695 fixed-length code000001010011100101 variable-length code 010110011111011100

12. Hash Tables12 Encode: abc = 0.101.100 = 0101100 (just concatenate the codewords.) Decode: 001011101 = 0.0.101.1101 = aabe (use the (right)binary tree below:) a:4 5 b:13c:1 2 d:16e: 9 f:5 0 1 10 0 1486 142858 00 0 0 0 11 1 1 a:4 5 b:13c:1 2 d:16 e: 9 f:5 55 2530 14 10 0 01 0 0 0 0 11 1 1 Tree for the fixed length codeword Tree for variable- length codeword

13. Hash Tables13 Binary tree Every nonleaf node has two children. The fixed-length code in our example is not optimal. The total number of bits required to encode a file is f ( c ) : the frequency (number of occurrences) of c in the file d T (c): denote the depth of c ’ s leaf in the tree

14. Hash Tables14 Constructing an optimal code Formal definition of the problem: Input: a set of characters C={c 1, c 2, …, c n }, each c  C has frequency f[c]. Output: a binary tree representing codewords so that the total number of bits required for the file is minimized. Huffman proposed a greedy algorithm to solve the problem.

15. Hash Tables15 a:4 5 d:16e: 9 f:5b:13c:1 2 a:4 5 d:16 e: 9 f:5 14 01 b:13c:1 2 (a) (b)

16. Hash Tables16 a:4 5 d:16 e: 9 f:5 14 01 b:13c:1 2 25 01 a:4 5 b:13c:1 2 d:16 e: 9 f:5 2530 14 01 0 011 (c) (d)

17. Hash Tables17 a:4 5 b:13c:1 2 d:16 e: 9 f:5 55 2530 14 10 0 01 0 0 0 0 11 1 1 a:4 5 b:13c:1 2 d:16 e: 9 f:5 55 2530 14 01 0 0 0 11 1 (e) (f)

18. Hash Tables18 HUFFMAN(C) 1n:=|C| 2Q:=C 3for i:=1 to n-1 do 4z:=ALLOCATE_NODE() 5x:=left[z]:=EXTRACT_MIN(Q) 6y:=right[z]:=EXTRACT_MIN(Q) 7f[z]:=f[x]+f[y] 8INSERT(Q,z) 9return EXTRACT_MIN(Q)

19. Hash Tables19 The Huffman Algorithm This algorithm builds the tree T corresponding to the optimal code in a bottom-up manner. C is a set of n characters, and each character c in C is a character with a defined frequency f[c]. Q is a priority queue, keyed on f, used to identify the two least-frequent characters to merge together. The result of the merger is a new object (internal node) whose frequency is the sum of the two objects.

20. Hash Tables20 Time complexity Lines 4-8 are executed n-1 times. Each heap operation in Lines 4-8 takes O(lg n) time. Total time required is O(n lg n). Note: The details of heap operation will not be tested. Time complexity O(n lg n) should be remembered.

21. Hash Tables21 Another example: e:4a:6c:6b:9d:11 c:6b:9d:11 e:4a:6 10 01

22. Hash Tables22 d:11 e:4a:6 10 01 c:6b:9 15 0 1 c:6b:9 15 0 1 d:11 e:4a:6 10 01 21 01

23. Hash Tables23 c:6b:9 15 0 1 d:11 e:4a:6 10 01 21 01 36 01 Summary Huffman Code: Given a set of characters and frequency, you should be able to construct the binary tree for Huffman codes. Proofs for why this algorithm can give optimal solution are not required.