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Error Resilient LZ'77 Data Compression: Algorithms, Analysis, and Experiments

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KWY is a Computer Scientist, Algorithm Analysis and Design Expert, Bioinformatics Expert, Information Theory, Multimedia Compression Consultant, Computer Systems and Computer Communication Networks Consultant with world-class expertise in analysis and design of algorithms, bioinformatics, information theory, multimedia compression, random structures, performance evaluation, analytic combinatorics, discrete mathematics, pattern matching, stability problems in distributed systems, modeling of computer systems and computer communication networks, queueing theory, and operations research.

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Abstract
We propose a joint source-channel coding algorithm capable of correcting some errors in the popular Lempel-Ziv'77 scheme without introducing any measurable degradation in the compression performance. This can be achieved because the LZ'77 encoder does not completely eliminate the redundancy present in the input sequence. One source of redundancy can be observed when an LZ'77 phrase has multiple matches. In this case, LZ'77 can issue a pointer to any of those matches, and a particular choice carries some additional bits of information. We call a scheme with embedded redundant information the LZS'77 algorithm. We analyze the number of longest matches in such a scheme and prove that it follows the logarithmic series distribution with mean 1/h (plus some fluctuations), where h is the source entropy. Thus, the distribution associated with the number of redundant bits is well concentrated around its mean, a highly desirable property for error correction. These analytic results are proved by a combination of combinatorial, probabilistic and analytic methods (e.g., Mellin transform, depoissonization, combinatorics on words). In fact, we analyze LZS'77 by studying the multiplicity matching parameter in a suffix tree, which in turn is analyzed via comparison to its independent version, called trie. Finally, we present an algorithm in which a channel coder (e.g., Reed-Solomon coder) succinctly uses the inherent additional redundancy left by the LZS'77 encoder to detect and correct a limited number of errors. We call such a scheme the LZRS'77 algorithm. LZRS'77 is perfectly backward-compatible with LZ'77, that is, a file compressed with our error-resistant LZRS'77 can still be decompressed by a generic LZ'77 decoder.

Index Terms - Lempel-Ziv'77 scheme, multiple matches, joint source-channel coding, Reed-Solomon code, suffix trees, tries, Mellin transform, depoissonization, pattern matching, autocorrelation polynomial, combinatorics on words.

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Read other articles by this KKAI Associate:

Detecting Conserved Interaction Patterns in Biological Networks

A Framework for Self Manipulating Video Streams

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2D-Pattern Matching Image and Video Compression: Theory, Algorithms, and Experiments

Computer Scientist, Algorithm Analysis and Design Expert, Bioinformatics Expert, Information Theory, Multimedia Compression Consultant, Computer Systems and Computer Communication Networks Consultant, analysis and design of algorithms, bioinformatics, information theory, multimedia compression, random structures, performance evaluation, analytic combinatorics, discrete mathematics, pattern matching, stability problems in distributed systems, modeling of computer systems and computer communication networks, queueing theory, and operations research.
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