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1. CSE 326 Huffman coding Richard Anderson

2. Coding theory

Conversion, Encryption, Compression

Binary coding

Variable length coding

Code examples 000,001,010,011,100,101 1,01,001,0001,00001,000001 00,010,011,100,11,101 A B C D E F

3. Decode the following 11010010010101011 100100101010 Ambiguous Prefix code E 0 T 11 N 100 I 1010 S 1011 E 0 T 10 N 100 I 0111 S 1010

4. Prefix code

No prefix of a codeword is a codeword

Uniquely decodable

A 00 1 00 B 010 01 10 C 011 001 11 D 100 0001 0001 E 11 00001 11000 F 101 000001 101

5. Prefix codes and binary trees

Tree representation of prefix codes

A 00 B 010 C 0110 D 0111 E 10 F 11

6. Construct the tree for the following code E 0 T 11 N 100 I 1010 S 1011

7. Minimum length code

Average cost

Average leaf depth

Huffman tree tree with minimum weighted path length

C(T) weighted path length

8. Compute average leaf depth A 00 1/4 B 010 1/8 C 0110 1/16 D 0111 1/16 E 1 1/2

9. Huffman code algorithm

Derivation

Two rarest items will have the longest codewords

Codewords for rarest items differ only in the last bit

Idea: suppose the weights are with and the smallest weights

Start with an optimal code for and

Extend the codeword for to get codewords for and

10. Huffman code H = new Heap() for each w i T = new Tree(w i ) H.Insert(T) while H.Size() > 1 T 1 = H.DeleteMin() T 2 = H.DeleteMin() T 3 = Merge(T 1 , T 2 ) H.Insert(T 3 )

11. Example: Weights 4, 5, 6, 7, 11, 14, 21 4 5 6 7 11 14 21

12. Draw a Huffman tree for the following data values and show internal weights: 3, 5, 9, 14, 16, 35

13. Correctness proof

The most amazing induction proof

Induction on the number of code words

The Huffman algorithm finds an optimal code for n = 1

Suppose that the Huffman algorithm finds an optimal code for codes size n, now consider a code of size n + 1 . . .

14. Key lemma

Given a tree T, we can find a tree T, with the two minimum cost leaves as siblings, and C(T)

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