The LZ family

 

LZ77

    LZR LZSS LZB LZH – used by zip and unzip

LZ78

     LZW – Unix compress LZC – Unix compress LZT LZMW LZJLZFG

Overview of LZ family

 To demonstrate:   simple alphabet containing only two letters, a and b, and create a sample stream of text

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.

Sender : The Compressor

 Before compression, the pieces of text from the breaking-down process are indexed from 1 to n:

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.

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

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

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 (8 + 8 + 8) needed for the actual text aab.

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..

Lempel-Ziv applet

 See  http://www.cs.mcgill.ca/~cs251/OldCourses/1997/topic23/#JavaApplet

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 (257-4096) represent strings

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

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

Demonstrations

  Another animated LZ algorithm … http://www.data-compression.com/lempelziv.html

LZW encoder example

 compress the string BABAABAAA

LZW decoder

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

LZW decoder example

 decompress the string <66><65><256><257><65><26 0>

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

LZW advantages/disadvantages

  advantages  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

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)

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