Theoretical efficiency[ edit ] In the second of the two papers that introduced these algorithms they are analyzed as encoders defined by finite-state machines. A measure analogous to information entropy is developed for individual sequences as opposed to probabilistic ensembles. This measure gives a bound on the data compression ratio that can be achieved. It is then shown that there exist finite lossless encoders for every sequence that achieve this bound as the length of the sequence grows to infinity. In this sense an algorithm based on this scheme produces asymptotically optimal encodings. This result can be proven more directly, as for example in notes by Peter Shor.

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This web page attempts to discuss my implementation and the updates that have sporadically taken place since my original LZSS release. I also toyed with an in-memory implementation of the LZSS algorithm that preforms all encode and decode operations on arrays rather than files. This implementation might be useful to those developing on systems that do not include a file system. Information on downloading the source code for all of my LZSS implementations may be found here.

The rest of this page discusses the results of my efforts so far. Unlike Huffman coding, which attempts to reduce the average amount of bits required to represent a symbol, LZSS attempts to replace a string of symbols with a reference to a dictionary location for the same string. It is intended that the dictionary reference should be shorter than the string it replaces. The larger N, the longer it takes to search the whole dictionary for a match and the more bits will be required to store the offset into the dictionary.

Typically dictionaries contain an amount of symbols that can be represented by a whole power of 2. A symbol dictionary would require 9 bits to represent all possible offsets. If you need to use 9 bits, you might as well have a symbol dictionary so that you can have more entries.

Additional new symbols cause an equal number of the oldest symbols to slide out. Representing Strings As stated above, encoded strings are represented as an offset and a length.

Since the dictionary size is fixed at N the largest offset may be N - 1 , and the longest string matching a series of characters in the dictionary may be N characters. In the examples I have seen, N is typically or and the maximum length allowed for matching strings is typically between 10 and 20 characters.

LZ77 vs. Storer and Szymanski observed that individual unmatched symbols or matched strings of one or two symbols take up more space to encode than they do to leave uncoded.

For example, encoding a string from a dictionary of symbols, and allowing for matches of up to 15 symbols, will require 16 bits to store an offset and a length. A single character is only 8 bits. Using this method, encoding a single character doubles the required storage. Using the above example it now takes 9 bits for each uncoded symbol and 17 bits for each encoded string. Putting it All Together Based on the discussion above, encoding input requires the following steps: Step 1.

Initialize the dictionary to a known value. Step 2. Read an uncoded string that is the length of the maximum allowable match.

Step 3. Search for the longest matching string in the dictionary. Step 4. If a match is found greater than or equal to the minimum allowable match length: Write the encoded flag, then the offset and length to the encoded output. Otherwise, write the uncoded flag and the first uncoded symbol to the encoded output. Step 5. Shift a copy of the symbols written to the encoded output from the unencoded string to the dictionary. Step 6. Read a number of symbols from the uncoded input equal to the number of symbols written in Step 4.

Step 7. Repeat from Step 3, until all the entire input has been encoded. The encoding process requires that a the dictionary is searched for matches to the string to be encoding. Decoding an offset and length combination only requires going to a dictionary offset and copying the specified number of symbols. No searching is required. Decoding input requires the following steps: Step 1. If the flag indicates an encoded string: Read the encoded length and offset, then copy the specified number of symbols from the dictionary to the decoded output.

Otherwise, read the next character and write it to the decoded output. Shift a copy of the symbols written to the decoded output into the dictionary.

Repeat from Step 2, until all the entire input has been decoded. Encoded strings should not be longer than two bytes 16 bits.

My version 0. All of my versions use codes that are integral bytes, and each of my newer versions are derived from my earlier versions so there is a lot of common design. What follows is a discussion about how these goals were accomplished. The Dictionary The dictionary size directly effects: The time to search the entire dictionary The size of the encoded offset into the dictionary The likelihood of a string matching one that already exists in the dictionary I chose to implement my dictionary as a character cyclic buffer sliding window.

I chose characters, because others have achieved good results with dictionaries of this size, and I can encode offsets of 0 to in 12 bits. If I encode the offset and length of a string in a 16 bit word, that leaves me with 4 bits for the length.

With 4 bits, I can encode lengths of 0 through However, as Storer and Szymanski observed, it only takes 1 byte to write out characters that match dictionary strings 0 or 1 character long, and 2 bytes to write out characters that match dictionary strings 2 characters long.

By not encoding strings that offer less than one byte of savings, the minimum encoded length is three characters. Since the minimum encoded length is three, I am able to add three to the 4 bit value stored in the length field, resulting in encoded string lengths of 3 to So 18 is the maximum string length encoded by this implementation. Similarly writing an encoded string would require 17 bits, 1 bit for the flag and 16 bits for the code.

While I was studying the algorithm, I came across some implementations that stored encoded flags in groups of eight followed by the characters or encoded strings that they represent. This technique also makes the EOF clear. By processing only bytes, there are no spare bits, os the EOF of the encoded dats is actually the EOF of the encoded file.

After writing my version 0. Finding Matches in the Dictionary As stated above, one of my goals was to provide an LZSS implementation that is easy to read and easy to debug.

So, to keep things simple, my first implementation used a brute force sequential search to match the string to be encoded to strings in the dictionary.



Gakasa The logic is pretty simple, but may require a number of comparisons. KMP is smart enough to skip ahead and resume by comparing string[0] to dictionary[3]which happens to be a match in this example. The first search I implemented was a sequential search. One approach to ensuring that the first character of the string being encoded matches the dictionary entry it is being compared to is to keep a list of all dictionary entries that start with a character for each character in the alphabet. The source code implementing a linked list search is contained in the version 0. Accessed on July 13, The sequential search algorithm moves through the dictionary one character at a time, checking for matches to lzsx first character in the string to be encoded. If a match is found greater than or algofithm to the minimum allowable match length:.


LZ77 and LZ78

Fenrir The sequential search algorithm moves through the dictionary one character at a time, checking for matches to the first character in the string to be encoded. The binary tree may then be used as a search optimization. Linked lists are a natural choice for implementing such dynamic lists. In my implementation, all pointers left, right, and parent are all unsigned int indices into the sliding window dictionary. So, to keep things simple, my first implementation used a brute force sequential search to match the string to be encoded to strings in the dictionary. Compression formats Compression software codecs.


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