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On the average redundancy rate of the Lempel-Ziv code

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2 Author(s)
Louchard, G. ; Lab. d''Inf. Theorique, Univ. Libre de Bruxelles, Belgium ; Szpankowski, W.

In this paper, we settle a long-standing open problem concerning the average redundancy rn of the Lempel-Ziv'78 (LZ78) code. We prove that for a memoryless source the average redundancy rate attains asymptotically Ern=(A+δ(n))/log n+ O(log log n/log2 n), where A is an explicitly given constant that depends on source characteristics, and δ(x) is a fluctuating function with a small amplitude. We also derive the leading term for the kth moment of the number of phrases. We conclude by conjecturing a precise formula on the expected redundancy for a Markovian source. The main result of this paper is a consequence of the second-order properties of the Lempel-Ziv algorithm obtained by Jacquet and Szpankowski (1995). These findings have been established by analytical techniques of the precise analysis of algorithms. We give a brief survey of these results since they are interesting in their own right, and shed some light on the probabilistic behavior of pattern matching based data compression

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Information Theory, IEEE Transactions on  (Volume:43 ,  Issue: 1 )