LZ77 Encoding Algorithm

Transcription

1 Components of LZ77 General Strategy: See if the current character (or sequence of characters) in the input string has already occurred earlier in the input data. If it has, then output a pointer to the earlier occurrence. LZ77 is implemented by using a sliding window that is divided into two parts: - the search buffer which represents the most recently encoded characters. It is an implicit dictionary, so we don't have to roll the dictionary into the compressed data - a lookahead buffer which contains characters yet to be encoded. This lookahead buffer starts where the search buffer ends, and, at times during the algorithm, the search buffer extends into the lookahead buffer. November 16 2

2 LZ77 Encoding Algorithm 1) Start with the first character of the lookahead buffer 2) Find the longest match in the window for the lookahead buffer. That is, find the longest string in the search buffer that matches a recurring pattern at the beginning of the lookahead buffer. (If possible take into account that end of the search buffer in the beginning of the lookahead buffer.) 3) Output the triplet <O, L, C> where: O: is the backward offset from beginning of lookahead buffer L: is the length of the match C: is the code for the first character in the lookahead buffer that didn't match; 4) If not done, move the sliding window L+1 characters forward and return to step 2. November 16 3

3 LZ77: An Example Assume that LZ77 has a sliding window of size 20 with lookahead buffer size of 10, and search buffer size of 10. Use it to encode: she sells sea shells by the seashore $ denotes the end of input and _ denotes the space. Use C(x) to denote the encoding of character x. This encoding might well be done using another compression technique. LZ77: Remarks The search buffer is generally thousands of bytes long, while the lookahead buffer is tens of bytes long. The encoding can be quite time consuming, due to the large number of pattern-matching comparisons performed on the characters in the lookahead and search buffers. LZ77 assumes (hopes, prays) that patterns in the input string occur close together. Decoding? An improvement is achieved by eliminating the need of the third component LZ78 and LZW. November 16 4

4 In the LZ77 data compression algorithm, parameters determining the efficiency of the algorithm are the size of the file being compressed (F), the size of the search buffer (S), the size of the lookahead buffer (L), and the number of characters in the alphabet of the data being compressed (C). Assume F is an order of magnitude bigger than C -- otherwise the worst case part of this question is too easy. Provide an example of the best case execution of the LZ77 algorithm. Here "best case" refers to the best possible compression, not the fastest run time. For this best case express the size of the encoded (compressed) data in big-o terms using the four parameters F, S, L, and C. For this best case express the run time of the algorithm in big-o terms using the four parameters F, S, L, and C. Provide an example of the worst case execution of the LZ77 algorithm. Here "worst case" refers to the worst possible compression, not the slowest run time. For this worst case express the size of the encoded (compressed) data in big-o terms using the four parameters F, S, L, and C. For this worst case express the run time of the algorithm in big-o terms using the four parameters F, S, L, and C. November 16 5

5 Where is the LZ77 dictionary? What are its limitations? LZ78 Algorithms Create a dictionary of the phrases that occur in the input data. When the encoder encounters a phrase already present in the dictionary, the index number of the phrase in the dictionary is used as code. Components of LZ78 LZ78 keeps a dictionary of previously encountered strings. The encoder outputs tokens of the form: <pointer to string P, code of symbol C> where: - P denotes a string from the dictionary. - C denotes the input character currently being processed. November 16 6

6 LZ78 Encoding Algorithm 1) At the start, the dictionary and P are empty; 2) C = next input character to be processed; 3) Is the string P+C in the dictionary? (a) if it is, P = P+C (concatenate C with P); (b) if not, i) output the token <pointer to string P, code of C>; ii) add the string P+C to the dictionary; iii) P = empty; (4) Are there more input characters to be processed? (a) if yes, return to step 2; (b) if not: i) if P is not empty, output the token <pointer to P, code of C>; ii) END. LZ78: An Example Use LZ78 to encode: sir sid eastman easily teases sea sick seals The symbol _ denotes the space and $ end of file. "x" will denote the encoding of character x. At the beginning the dictionary is empty. It starts with the null string at position zero. November 16 7

7 After the First 12 Encoding Steps sir sid easeman easily teases sea sick seals$ Dictionary Output Position String Token 0 null 1 s <0, "s"> 2 i <0, "i"> 3 r <0, "r"> 4 - <0, "-"> 5 si <1, "i"> 6 d <0, "d"> 7 -e <4, "e"> 8 a <0, "a"> 9 st <1, "t"> 10 m <0, "m"> 11 an <8, "n"> 12 -ea <7, "a"> November 16 8

8 Final Dictionary and Encoding (Continued) sir sid eastman easily teases sea sick seals$ 13 sil <5,"l"> 14 y <0, "y"> 15 -t <4, "t"> 16 e <0, "e"> 17 as <8, "s"> 18 es <16, "s"> 19 -s <4, "s"> 20 ea <16, "a"> 21 -si <19, "i"> 22 c <0, "c"> 23 k <0, "k"> 24 -se <19, "e"> 25 al <8, "l"> 26 s$ <l, "$"> November 16 9