On Certain Pathwise Properties of the Sliding Window Lempel-Ziv Algorithm

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We derive a number of almost sure results related to the sliding window Lempel-Ziv (SWLZ) algorithm. A principal result is a pathwise lower bound to the redundancy which is off by a factor of two from the main term in the lower bound of Wyner [1] and Yang and Kieffer [2], which hold inthe expected sense for the Fixed Database Lempel Ziv algorithm, for different definitions of the compression ratio. Other results include a relatively weak upper bound on the redundancy and reasonably tight upper and lower bounds for the number of bits spent on the encoding of the phrase lengths. The main theme in deriving these results is a simple phrase length thresholding technique that traces its roots back to the work of Wyner and Ziv [3]; a significant aspect the present work can be viewed as exploring the extent to which said thresholding idea can give more detailed information about the properties of SWLZ. Another aspect of the present work studies the asymptotic behavior of the ratio of the number of phrases to the length of the parsed string for any finite sliding window size; in here we exploit results of Kieffer and Rahe [4]. In all cases it is assumed that the source is stationary and that in the most restrictive case it is an irreducible and aperiodic Markov chain; some of the results hold for sources that have exponential rates for entropy and more generally for the ergodic setting.

By: Luis Alfonso Lastras-Montaño

Published in: IEEE Transactions on Information Theory, volume 52, (no 12), pages 5267-83 in 2006


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