Data Compression Hae-sun Jung CS146 Dr. Sin-Min Lee Spring 2004

1. Data Compression Hae-sun Jung CS146 Dr. Sin-Min Lee Spring 2004

3. Introduction • Compression is used to reduce the volume of information to be stored into storages or to reduce the communication bandwidth required for its transmission over the networks

5. Compression Principles • Entropy Encoding • Run-length encoding • Lossless & Independent of the type of source information • Used when the source information comprises long substrings of the same character or binary digit (string or bit pattern, # of occurrences), as FAX e.g) 000000011111111110000011……  0,7 1, 10, 0,5 1,2……  7,10,5,2……

6. Compression Principles • Entropy Encoding • Statistical encoding • Based on the probability of occurrence of a pattern • The more probable, the shorter codeword • “Prefix property”: a shorter codeword must not form the start of a longer codeword

7. Compression Principles • Huffman Encoding • Entropy, H: theoretical min. avg. # of bits that are required to transmit a particular stream H = -Σi=1nPi log2Pi where n: # of symbols, Pi: probability of symbol i • Efficiency, E=H/H’ where, H’ = avr. # of bits per codeword = Σi=1nNiPi Ni: # of bits of symbol i

8. E.g) symbols M(10), F(11), Y(010), N(011), 0(000), 1(001) with probabilities 0.25, 0.25, 0.125, 0.125, 0.125, 0.125 • H’ = Σi=16NiPi = (2(20.25) + 4(30.125)) = 2.5 bits/codeword • H = -Σi=16Pi log2Pi= - (2(0.25log20.25) + 4(0.125log20.125)) = 2.5 • E = H/H’ =100 % • 3-bit/codeword if we use fixed-length codewords for six symbols

9. Huffman Algorithm Method of construction for an encoding tree • Full Binary Tree Representation • Each edge of the tree has a value, (0 is the left child, 1 is the right child) • Data is at the leaves, not internal nodes • Result: encoding tree • “Variable-Length Encoding”

10. Huffman Algorithm • 1. Maintain a forest of trees • 2. Weight of tree = sum frequency of leaves • 3. For 0 to N-1 • Select two smallest weight trees • Form a new tree

11. Huffman coding • variable length code whose length is inversely proportional to that character’s frequency • must satisfy nonprefix property to be uniquely decodable • two pass algorithm • first pass accumulates the character frequency and generate codebook • second pass does compression with the codebook

12. Huffman coding • create codes by constructing a binary tree 1. consider all characters as free nodes 2. assign two free nodes with lowest frequency to a parent nodes with weights equal to sum of their frequencies 3. remove the two free nodes and add the newly created parent node to the list of free nodes 4. repeat step2 and 3 until there is one free node left. It becomes the root of tree

13. Right of binary tree :1 • Left of Binary tree :0 • Prefix (example) • e:”01”, b: “010” • “01” is prefix of “010” ==> “e0” • same frequency : need consistency of left or right

14. Example(64 data) • R K K K K K K K • K K K R R K K K • K K R R R R G G • K K B C C C R R • G G G M C B R R • B B B M Y B B R • G G G G G G G R • G R R R R G R R

15. Color frequency Huffman code • ================================= • R 19 00 • K 17 01 • G 14 10 • B 7 110 • C 4 1110 • M 2 11110 • Y 1 11111

17. 0 1 Root node 8 0 1 A 4 Branch node 0 1 2 B Leaf node D C Static Huffman Coding • Huffman (Code) Tree • Given : a number of symbols (or characters) and their relative probabilities in prior • Must hold “prefix property” among codes Symbol Occurrence A 4/8 B 2/8 C 1/8 D 1/8 Symbol Code A 1 B 01 C 001 D 000 41 + 22 + 13 + 13 = 14 bits are required to transmit “AAAABBCD” Prefix Property !

18. The end