1. The LZ family LZ77 LZ78 LZR LZSS LZB LZH – used by zip and unzip
LZW – Unix compress
LZC – Unix compress
LZT
LZMW
LZJLZFG

2. Overview of LZ family To demonstrate:
simple alphabet containing only two letters, a and b,
and create a sample stream of text

3. LZ family overview
Rule: Separate this stream of characters into pieces of text so that the shortest piece of data is the string of characters that we have not seen so far.

4. Sender : The Compressor
Before compression, the pieces of text from the breaking-down process are indexed from 1 to n:

5. LZ indices are used to number the pieces of data.
The empty string (start of text) has index 0.
The piece indexed by 1 is a. Thus a, together with the initial string, must be numbered Oa.
String 2, aa, will be numbered 1a, because it contains a, whose index is 1, and the new character a.

6. LZ the process of renaming pieces of text starts to pay off.
Small integers replace what were once long strings of characters.
can now throw away our old stream of text and send the encoded information to the receiver

7. Bit Representation of Coded Information
Now, want to calculate num bits needed
each chunk is an int and a letter
num bits depends on size of table permitted in the dictionary
every character will occupy 8 bits because it will be represented in US ASCII format

8. Compression good?
in a long string of text, the number of bits needed to transmit the coded information is small compared to the actual length of the text.
example: 12 bits to transmit the code 2b instead of 24 bits ( ) needed for the actual text aab.

9. Receiver: The Decompressor (Implementation
receiver knows exactly where boundaries are, so no problem in reconstructing the stream of text.
Preferable to decompress the file in one pass; otherwise, we will encounter a problem with temporary storage..

10. Lempel-Ziv applet
See

11. Lempel Ziv Welsch (LZW)
previous methods worked only on characters
LZW works by encoding strings
some strings are replaced by a single codeword
for now assume codeword is fixed (12 bits)
for 8 bit characters, first 256 (or less) entries in table are reserved for the characters
rest of table ( ) represent strings

12. LZW compression
trick is that string-to-codeword mapping is created dynamically by the encoder
also recreated dynamically by the decoder
need not pass the code table between the two
is a lossless compression algorithm
degree of compression hard to predict
depends on data, but gets better as codeword table contains more strings

13. LZW encoder Initialize table with single character strings
STRING = first input character
WHILE not end of input stream
CHARACTER = next input character
IF STRING + CHARACTER is in the string table
STRING = STRING + CHARACTER
ELSE
Output the code for STRING
Add STRING + CHARACTER to the string table
STRING = CHARACTER
END WHILE
Output code for string

14. Demonstrations Another animated LZ algorithm …

15. LZW encoder example
compress the string BABAABAAA

16. LZW decoder

17. Lempel-Ziv compression
a lossless compression algorithm
All encodings have the same length
But may represent more than one character
Uses a “dictionary” approach – keeps track of characters and character strings already encountered

18. LZW decoder example
decompress the string <66><65><256><257><65><260>

19. LZW Issues compression better as the code table grows
what happens when all 4096 locations in string table are used?
A number of options, but encoder and decoder must agree to do the same thing
do not add any more entries to table (as is)
clear codeword table and start again
clear codeword table and start again with larger table/longer codewords (GIF format)

20. LZW advantages/disadvantages
simple, fast and good compression
can do compression in one pass
dynamic codeword table built for each file
decompression recreates the codeword table so it does not need to be passed
disadvantages
not the optimum compression ratio
actual compression hard to predict

21. Entropy methods all previous methods are lossless and entropy based
lossless methods are essential for computer data (zip, gnuzip, etc.)
combination of run length encoding/huffman is a standard tool
are often used as a subroutine by other lossy methods (Jpeg, Mpeg)

22. Lempel-Ziv compression
a lossless compression algorithm
All encodings have the same length
But may represent more than one character
Uses a “dictionary” approach – keeps track of characters and character strings already encountered