Top Document: comp.compression Frequently Asked Questions (part 2/3) Previous Document: [72] What is wavelet theory? Next Document: [74] Introduction to JBIG See reader questions & answers on this topic!  Help others by sharing your knowledge This question can be understood in two different ways: (a) For a given compressor/decompressor, what is the best possible lossless compression for an arbitrary string (byte sequence) given as input? (b) For a given string, what is the best possible lossless compressor/decompressor? For case (a), the question is generally meaningless, because a specific compressor may compress one very large input file down to a single bit, and enlarge all other files by only one bit. There is no lossless compressor that is guaranteed to compress all possible input files. If it compresses some files, then it must enlarge some others. This can be proven by a simple counting argument (see item 9). In case (a), the size of the decompressor is not taken into account for the determination of the compression ratio since the decompressor is fixed and it may decompress an arbitrary number of files of arbitrary length. For case (b), it is of course necessary to take into account the size of the decompressor. The problem may be restated as "What is the shortest program P which, when executed, produces the string S?". The size of this program is known as the Kolmogorov complexity of the string S. Some (actually most) strings are not compressible at all, by any program: the smallest representation of the string is the string itself. On the other hand, the output of a pseudorandom number generator can be extremely compressible, since it is sufficient to know the parameters and seed of the generator to reproduce an arbitrary long sequence. References: "An Introduction to Kolmogorov Complexity and its Applications", Ming Li and Paul Vitanyi, 2nd edition, SpringerVerlag, ISBN 0387948686 http://www.cwi.nl/~paulv/kolmogorov.html If you don't want to read a whole book, I recommend the excellent lecture "Randomness & Complexity in Pure Mathematics" by G. J. Chaitin: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ijbc.html The decimal and binary expansions of Chaitin's number Omega are examples of uncompressible strings. There are more papers on http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ User Contributions:Comment about this article, ask questions, or add new information about this topic:Top Document: comp.compression Frequently Asked Questions (part 2/3) Previous Document: [72] What is wavelet theory? Next Document: [74] Introduction to JBIG Part1  Part2  Part3  Single Page [ Usenet FAQs  Web FAQs  Documents  RFC Index ] Send corrections/additions to the FAQ Maintainer: jloup@gzip.OmitThis.org
Last Update March 27 2014 @ 02:11 PM
