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Average redundancy rate of the Lempel-Ziv code

par Louchard, Guy ;Szpankowski, Wojciech
Référence Data Compression Conference Proceedings, page (92-101)
Publication Publié, 1996

Article révisé par les pairs

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Résumé :

It was conjectured that the average redundancy rate, rn, for the Lempel-Ziv code (LZ78) is Θ(log log n/log n) where n is the length of the database sequence. However, it was also known that for infinitely many n the redundancy rn is bounded from the below by 2/ log n. In this paper we settle the above conjecture in the negative by proving 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 the source characteristics, and δ(x) is a fluctuating function. This result is a consequence of recently established second-order properties for the number of phrases in the Lempel-Ziv algorithm. We also derive the leading term for the kth moment of the number of phrases. Finally, in concluding remarks we discuss generalized Lempel-Ziv codes for which the average redundancy rates are computed and compared with the original Lempel-Ziv code.