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The LempelZivWelch (LZW) Algorithm Tom Magerlein May 10, 2017 - PowerPoint PPT Presentation

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The LempelZivWelch (LZW) Algorithm Tom Magerlein May 10, 2017 Asked during Bens presentation on Huffman coding: Can we replace common sequences of characters, like the, with single codes? Asked during Bens presentation on

  1. The Lempel–Ziv–Welch (LZW) Algorithm Tom Magerlein May 10, 2017

  2. Asked during Ben’s presentation on Huffman coding: Can we replace common sequences of characters, like “the”, with single codes?

  3. Asked during Ben’s presentation on Huffman coding: Can we replace common sequences of characters, like “the”, with single codes? Yes!

  4. Origins ◮ 1977-78: Lempel and Ziv introduce LZ77 and LZ78 ◮ LZ77: Replace previously seen sequences of characters with references to previous appearances ◮ LZ78: Instead of referencing earlier parts of file directly, builds a dictionary of previously-seen symbol sequences ◮ 1983: Sperry Corp. (later Unisys) files patent on original LZW implementation ◮ 1984: Welch publishes “A Technique for High-Performance Data Compression”, describing the LZW algorithm

  5. Origins

  6. Uses and Patent Troubles ◮ Saw use in some compression utilities, but most notable use was in CompuServe’s GIF image format, introduced in 1987 ◮ In 1993/4, Unisys discovers use of LZW in GIF format, attempts to claim licensing fees from software that handles GIF images ◮ Leads to development of the patent-unencumbered PNG format and the widespread use of the DEFLATE compression algorithm, as well as use of the GIF format without compression ◮ Patent expired in 2003, but still not widely used except in GIF

  7. LZ77: Overview ◮ Replaces previously seen data segments with a reference to where they last occurred, as a pair indicating offset and sequence length ◮ Compressor keeps an output history (typically a few kilobytes), called the “sliding window”, and a lookahead buffer ◮ Algorithm ◮ Find longest prefix of data in lookahead buffer which occurs in sliding window ◮ If such a prefix exists, and it would save space to do so, output reference to its last occurrence; otherwise output first unit of data as a literal

  8. LZ77: A Short Example A B C B A B C B C B C

  9. LZ77: A Short Example A B C B A B C B C B C A B C B

  10. LZ77: A Short Example A B C B A B C B C B C A B C B A B C length 3

  11. LZ77: A Short Example A B C B A B C B C B C length 4 A B C B A B C B C B C length 3

  12. LZ78: Overview ◮ Replaces LZ77 sliding window with a dictionary, and backreferences with codes representing entries in the dictionary ◮ Compressor, decompressor agree on rules to build dictionary, so it does not need to be stored with compressed data ◮ Algorithm: ◮ Find longest prefix of lookahead buffer in current dictionary ◮ Output code for that prefix ◮ Output code for first character after prefix ◮ Add prefix followed by next character to dictionary, if dictionary is not full

  13. LZ78: Example A B C B C B A A B C A B C B B B B B B Dictionary A 0000 B 0001 C 0010 EOF 0011

  14. LZ78: Example A B C B C B A A B C A B C B B B B B B Dictionary A 0000 B 0001 C 0010 EOF 0011

  15. LZ78: Example A B C B C B A A B C A B C B B B B B B Dictionary A 0000 B 0001 C 0010 EOF 0011

  16. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100

  17. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100

  18. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100

  19. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101

  20. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101

  21. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101

  22. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101

  23. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110

  24. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110

  25. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110

  26. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110

  27. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  28. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  29. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  30. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  31. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] Dictionary A 0000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  32. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] Dictionary A 0000 ABCB 1000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  33. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] Dictionary A 0000 ABCB 1000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  34. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] Dictionary A 0000 ABCB 1000 B 0001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  35. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  36. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  37. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  38. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  39. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] [BB] [B] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 BBB 1010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  40. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] [BB] [B] [ EOF ] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 BBB 1010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  41. LZ78: Example A B C B C B A A B C A B C B B B B B B [A] [B] [C] [B] [CB] [A] [AB] [C] [ABC] [B] [B] [B] [BB] [B] [ EOF ] Dictionary A 0000 ABCB 1000 B 0001 BB 1001 C 0010 BBB 1010 EOF 0011 AB 0100 CB 0101 CBA 0110 ABC 0111

  42. LZW: Overview ◮ Extends LZ78 to eliminate the requirement that the symbol at the end of a new dictionary entry be emitted as a literal, instead using it as first symbol of next prefix ◮ Now possible for decompressor to encounter codes before they are added to its dictionary:

  43. LZW: Overview ◮ Extends LZ78 to eliminate the requirement that the symbol at the end of a new dictionary entry be emitted as a literal, instead using it as first symbol of next prefix ◮ Now possible for decompressor to encounter codes before they are added to its dictionary: AAA → [A][AA] ◮ Unknown code must have been added to dictionary after encoding previously received sequence; must therefore code for the previously received sequence followed by one more character

  44. LZW: Overview ◮ Extends LZ78 to eliminate the requirement that the symbol at the end of a new dictionary entry be emitted as a literal, instead using it as first symbol of next prefix ◮ Now possible for decompressor to encounter codes before they are added to its dictionary: AAA → [A][AA] ◮ Unknown code must have been added to dictionary after encoding previously received sequence; must therefore code for the previously received sequence followed by one more character ◮ Last character of sequence must be same as first, since the new dictionary entry was last sequence followed by the last sequence again, followed by that character

  45. LZW: Example A B C B C B A A B C A B C B B B B B B Dictionary A 0000 B 0001 C 0010 EOF 0011

  46. LZW: Example A B C B C B A A B C A B C B B B B B B Dictionary A 0000 B 0001 C 0010 EOF 0011

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