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

We propose a joint source-channel coding algorithm capable of correcting some errors in the popular Lempel-Ziv´77 (LZ´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 (RS) 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