1. 1 Huffman Codes

2. 2 ASCII use same size encoding for all characters. Variable length codes can produce shorter messages than fixed length codes Huffman Codes constructs an optimal variable bit length code for an alphabet with known frequency probability for each character

3. Average Length for a Character Set ACDDE probability0.20.1 0.150.45 length23441 Expected length = 0.2*2 + 0.1*3+0.1*4+0.15*4+0.45*1

4. 4 Immediate Decodability Prefix codes: No valid code symbol is a prefix of another valid code symbol. – Codes that are immediately decodable!

5. 5 Huffman's code Optimal Codes – For a given character set with known frequency probability – Immediately decodable. – Average message length for a large number of messages is minimal

6. 6 Huffman's Algorithm Huffman_tree_node: ch, freq, *left and *right Initialize a list of Huffman_tree_node – One node for each character containing the character and the frequency, left, right is NULL. While there is more than one node in the list: – Find two nodes with minimal frequency. – Remove those nodes from the list, make a new node, the new node’s left and right subtrees are the nodes just removed, the new node frequency is the sum of left and right nodes’ frequency – Label the arc to the left subtree with 0. – Label the arc to the right subtree with 1. – Add the new node to the list.

7. Example ABCDE probability0.20.1 0.150.45 BCDAE 0.1 NULL 0.1 NULL 0.15 NULL 0.2 NULL 0.45 NULL Remove two nodes Band C which minimal freq, Make a new Huffman_tree_node: BC: left =B, right –C freq=0.1+0.1=0.2, put the new node into the list, next select D A, visualize the tree, Keep doing this until there is no more than 1 node in the list DABCE 0.15 NULL 0.2 NULL 0.2 B C 0.45 NULL

8. Huffm an code A 011 B 000 C 001 D 010 E1E1 ABCDE probability0.20.1 0.150.45 Example Result

9. 9 Huffman Decoding Algorithm HuffmanDecode(Huffman_Tree ht, String s), Initialize pointer p to the root of Huffman tree. While end of message string not reached: move p to the left child if next bit is 0 otherwise move p to the right child If p points to a leaf Display the character at that leaf. Reset p to the root of the Huffman tree.

10. 10 Huffman Decoding Algorithm Decoding 0001011010 Remember that Huffman code is immediately decodable: B E A D

11. 11 Implementing a Huffman Code Program class Huffman_tree_node { //functions to set ch, freq, left and right char ch; double freq; Huffman_tree_node *left, *right; }

12. 12 Huffman_Tree class Huffman_Tree { public: Huffman_Tree(void); ~Huffman_Tree(void) {}; // Add a single node tree to the list. void Add(char c, double frequency); Make_Huffman_code_Tree(void) private: list node_list; };

13. 13 main.cpp Huffman_Tree huffman_tree; int main(void) { cout << "This is the Huffman code program.\n\n"; huffman_tree.Add('a',0.2); huffman_tree.Add('b',0.1); huffman_tree.Add('c',0.1); huffman_tree.Add('d',0.15); huffman_tree.Add('e',0.45); }

14. 14 Implementing Huffman’s Algorithm function Make_Huffman_code_Tree Repeatedly – Sort the list of trees by frequency – Remove the first two trees – Create a new node with these trees as subtrees. Frequency is sum of their frequencies – Add the new node to the list. Continue until there is only one node on the list.

15. 15 Huffman_Tree.cpp function Make_Huffman_code_Tree () while (node_list.size() > 1) { Huffman_tree_node* cf1 = new Huffman_tree_node(node_list.front()); node_list.pop_front(); Huffman_tree_node* cf2 = new Huffman_tree_node(node_list.front()); node_list.pop_front(); Huffman_tree_node cf3(0, cf1->Freq()+cf2->Freq(), cf1, cf2); node_list.push_back(cf3); node_list.sort(); }