A suboptimal lossy data compression based on approximate pattern matching

• Authors: T. Luczak;W. Szpankowski
• Venue: IEEE Transactions on Information Theory
• Year: 2006

A practical suboptimal (variable source coding) algorithm for lossy data compression is presented. This scheme is based on approximate string matching, and it naturally extends the lossless Lempel-Ziv (1977) data compression scheme. Among others we consider the typical length of an approximately repeated pattern within the first n positions of a stationary mixing sequence where D percent of mismatches is allowed. We prove that there exists a constant r0(D) such that the length of such an approximately repeated pattern converges in probability to 1/r0(D) log n (pr.) but it almost surely oscillates between 1/r-∞(D) log n and 2/r1(D) log n, where r -∞(D)>r0(D)>r1(D)/2 are some constants. These constants are natural generalizations of Renyi entropies to the lossy environment. More importantly, we show that the compression ratio of a lossy data compression scheme based on such an approximate pattern matching is asymptotically equal to r0(D). We also establish the asymptotic behavior of the so-called approximate waiting time Nl which is defined as the time until a pattern of length C repeats approximately for the first time. We prove that log Nl/l→r0(D) (pr.) as l→∞. In general, r0(D)>R(D) where R(D) is the rate distortion function. Thus for stationary mixing sequences we settle in the negative the problem investigated by Steinberg and Gutman by showing that a lossy extension of the Wyner-Ziv (1989) scheme cannot be optimal